FORMALIZED MATHEMATICS Volume 11, Number 1, 2003 University of Białystok

Subspaces and Cosets of Subspace of Real Unitary Space

Noboru Endou Gifu National College of Technology Takashi Mitsuishi Miyagi University

Yasunari Shidama Shinshu University Nagano

Summary. In this article, subspace and the coset of subspace of real unitary space are defined. And we discuss some of their fundamental properties.

MML Identifier: RUSUB 1.

The articles [6], [3], [10], [7], [1], [11], [2], [5], [9], [8], and [4] provide the notation and terminology for this paper.

1. Definition and Axioms of the Subspace of Real Unitary Space

Let V be a real unitary space. A real unitary space is said to be a subspace of V if it satisfies the conditions (Def. 1).

(Def. 1)(i) The carrier of it ⊆ the carrier of V, (ii) the zero of it = the zero of V, (iii) the addition of it = (the addition of V)↾[: the carrier of it, the carrier

of it :], (iv) the external multiplication of it = (the external multiplication of

V)↾[: ℝ, the carrier of it :], and

(v) the scalar product of it = (the scalar product of V)↾[: the carrier of it, the carrier of it :].

We now state a number of propositions:

(1) Let V be a real unitary space, W_{1}, W_{2} be subspaces of V, and x be a set. If x ∈ W_{1} and W_{1} is a subspace of W_{2}, then x ∈ W_{2}.

(2) For every real unitary space V and for every subspace W of V and for every set x such that x ∈ W holds x ∈ V.

(3) For every real unitary space V and for every subspace W of V holds every vector of W is a vector of V.

(4) For every real unitary space V and for every subspace W of V holds 0_{W} = 0_{V} .

(5) For every real unitary space V and for all subspaces W_{1}, W_{2} of V holds 0_{(W_{1})} = 0_{(W_{2})}.

(6) Let V be a real unitary space, W be a subspace of V, u, v be vectors of V, and w_{1}, w_{2} be vectors of W. If w_{1} = v and w_{2} = u, then w_{1}+w_{2} = v+u.

(7) Let V be a real unitary space, W be a subspace of V, v be a vector of V, w be a vector of W, and a be a real number. If w = v, then a · w = a · v.

(8) Let V be a real unitary space, W be a subspace of V, v_{1}, v_{2} be vectors of V, and w_{1}, w_{2} be vectors of W. If w_{1} = v_{1} and w_{2} = v_{2}, then (w_{1}|w_{2}) = (v_{1}|v_{2}).

(9) Let V be a real unitary space, W be a subspace of V, v be a vector of V, and w be a vector of W. If w = v, then − v = − w.

(10) Let V be a real unitary space, W be a subspace of V, u, v be vectors of V, and w_{1}, w_{2} be vectors of W. If w_{1} = v and w_{2} = u, then w_{1}−w_{2} = v−u.

(11) For every real unitary space V and for every subspace W of V holds 0_{V} ∈ W.

(12) For every real unitary space V and for all subspaces W_{1}, W_{2} of V holds 0_{(W_{1})} ∈ W_{2}.

(13) For every real unitary space V and for every subspace W of V holds 0_{W} ∈ V.

(14) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. If u ∈ W and v ∈ W, then u + v ∈ W.

(15) Let V be a real unitary space, W be a subspace of V, v be a vector of V, and a be a real number. If v ∈ W, then a · v ∈ W.

(16) For every real unitary space V and for every subspace W of V and for every vector v of V such that v ∈ W holds − v ∈ W.

(17) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. If u ∈ W and v ∈ W, then u − v ∈ W.

(18) Let V be a real unitary space, V_{1} be a subset of the carrier of V, D be a non empty set, d_{1} be an element of D, A be a binary operation on D, M be a function from [: ℝ, D:] into D, and S be a function from [: D, D:] into ℝ. Suppose that (i) V_{1} = D, (ii) d_{1} = 0_{V} , (iii) A = (the addition of V)↾[: V_{1}, V_{1} :], (iv) M = (the external multiplication of V)↾[: ℝ, V_{1} :], and

(v) S = (the scalar product of V)↾[: V_{1}, V_{1} :]. Then ⟨D, d_{1}, A, M, S⟩ is a subspace of V.

(19) Every real unitary space V is a subspace of V.

(20) For all strict real unitary spaces V, X such that V is a subspace of X and X is a subspace of V holds V = X.

(21) Let V, X, Y be real unitary spaces. Suppose V is a subspace of X and X is a subspace of Y. Then V is a subspace of Y.

(22) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Suppose the carrier of W_{1} ⊆ the carrier of W_{2}. Then W_{1} is a subspace of W_{2}.

(23) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Suppose that for every vector v of V such that v ∈ W_{1} holds v ∈ W_{2}. Then W_{1} is a subspace of W_{2}.

Let V be a real unitary space. Observe that there exists a subspace of V which is strict.

Next we state several propositions:

(24) Let V be a real unitary space and W_{1}, W_{2} be strict subspaces of V. If the carrier of W_{1} = the carrier of W_{2}, then W_{1} = W_{2}.

(25) Let V be a real unitary space and W_{1}, W_{2} be strict subspaces of V. If for every vector v of V holds v ∈ W_{1} iff v ∈ W_{2}, then W_{1} = W_{2}.

(26) Let V be a strict real unitary space and W be a strict subspace of V. If the carrier of W = the carrier of V, then W = V.

(27) Let V be a strict real unitary space and W be a strict subspace of V. If for every vector v of V holds v ∈ W iff v ∈ V, then W = V.

(28) Let V be a real unitary space, W be a subspace of V, and V_{1} be a subset of the carrier of V. If the carrier of W = V_{1}, then V_{1} is linearly closed.

(29) Let V be a real unitary space, W be a subspace of V, and V_{1} be a subset of the carrier of V. Suppose V_{1} ≠ ∅ and V_{1} is linearly closed. Then there exists a strict subspace W of V such that V_{1} = the carrier of W.

2. Definition of Zero Subspace and Improper Subspace of Real Unitary Space

Let V be a real unitary space. The functor 0_{V} yields a strict subspace of V and is defined by:

(Def. 2) The carrier of 0_{V} = {0_{V} }.

Let V be a real unitary space. The functor Ω_{V} yielding a strict subspace of V is defined by:

(Def. 3) Ω_{V} = the unitary space structure of V.

3. Theorems of Zero Subspace and Improper Subspace

Next we state several propositions:

(30) For every real unitary space V and for every subspace W of V holds 0_{W} = 0_{V} .

(31) For every real unitary space V and for all subspaces W_{1}, W_{2} of V holds 0_{(W_{1})} = 0_{(W_{2})}.

(32) For every real unitary space V and for every subspace W of V holds 0_{W} is a subspace of V.

(33) For every real unitary space V and for every subspace W of V holds 0_{V} is a subspace of W.

(34) For every real unitary space V and for all subspaces W_{1}, W_{2} of V holds 0_{(W_{1})} is a subspace of W_{2}.

(35) Every strict real unitary space V is a subspace of Ω_{V}.

4. The Cosets of Subspace of Real Unitary Space

Let V be a real unitary space, let v be a vector of V, and let W be a subspace of V. The functor v + W yields a subset of the carrier of V and is defined as follows:

(Def. 4) v + W = {v + u; u ranges over vectors of V: u ∈ W }.

Let V be a real unitary space and let W be a subspace of V. A subset of the carrier of V is said to be a coset of W if:

(Def. 5) There exists a vector v of V such that it = v + W.

5. Theorems of the Cosets

We now state a number of propositions:

(36) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. Then 0_{V} ∈ v + W if and only if v ∈ W.

(37) For every real unitary space V and for every subspace W of V and for every vector v of V holds v ∈ v + W.

(38) For every real unitary space V and for every subspace W of V holds 0_{V} + W = the carrier of W.

(39) For every real unitary space V and for every vector v of V holds v+0_{V} = {v}.

(40) For every real unitary space V and for every vector v of V holds v+Ω_{V} = the carrier of V.

(41) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. Then 0_{V} ∈ v + W if and only if v + W = the carrier of W.

(42) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. Then v ∈ W if and only if v + W = the carrier of W.

(43) Let V be a real unitary space, W be a subspace of V, v be a vector of V, and a be a real number. If v ∈ W, then a · v + W = the carrier of W.

(44) Let V be a real unitary space, W be a subspace of V, v be a vector of V, and a be a real number. If a ≠ 0 and a · v + W = the carrier of W, then v ∈ W.

(45) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. Then v ∈ W if and only if − v + W = the carrier of W.

(46) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. Then u ∈ W if and only if v + W = v + u + W.

(47) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. Then u ∈ W if and only if v + W = (v − u) + W.

(48) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. Then v ∈ u + W if and only if u + W = v + W.

(49) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. Then v + W = − v + W if and only if v ∈ W.

(50) Let V be a real unitary space, W be a subspace of V, and u, v_{1}, v_{2} be vectors of V. If u ∈ v_{1} + W and u ∈ v_{2} + W, then v_{1} + W = v_{2} + W.

(51) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. If u ∈ v + W and u ∈ − v + W, then v ∈ W.

(52) Let V be a real unitary space, W be a subspace of V, v be a vector of V, and a be a real number. If a ≠ 1 and a · v ∈ v + W, then v ∈ W.

(53) Let V be a real unitary space, W be a subspace of V, v be a vector of V, and a be a real number. If v ∈ W, then a · v ∈ v + W.

(54) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. Then − v ∈ v + W if and only if v ∈ W.

(55) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. Then u + v ∈ v + W if and only if u ∈ W.

(56) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. Then v − u ∈ v + W if and only if u ∈ W.

(57) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. Then u ∈ v + W if and only if there exists a vector v_{1} of V such that

v_{1} ∈ W and u = v + v_{1}.

(58) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. Then u ∈ v + W if and only if there exists a vector v_{1} of V such that v_{1} ∈ W and u = v − v_{1}.

(59) Let V be a real unitary space, W be a subspace of V, and v_{1}, v_{2} be vectors of V. Then there exists a vector v of V such that v_{1} ∈ v + W and v_{2} ∈ v + W if and only if v_{1} − v_{2} ∈ W.

(60) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. If v+W = u+W, then there exists a vector v_{1} of V such that v_{1} ∈ W and v + v_{1} = u.

(61) Let V be a real unitary space, W be a subspace of V, and u, v be vectors of V. If v+W = u+W, then there exists a vector v_{1} of V such that v_{1} ∈ W and v − v_{1} = u.

(62) Let V be a real unitary space, W_{1}, W_{2} be strict subspaces of V, and v be a vector of V. Then v + W_{1} = v + W_{2} if and only if W_{1} = W_{2}.

(63) Let V be a real unitary space, W_{1}, W_{2} be strict subspaces of V, and u, v be vectors of V. If v + W_{1} = u + W_{2}, then W_{1} = W_{2}.

(64) Let V be a real unitary space, W be a subspace of V, and C be a coset of W. Then C is linearly closed if and only if C = the carrier of W.

(65) Let V be a real unitary space, W_{1}, W_{2} be strict subspaces of V, C_{1} be a coset of W_{1}, and C_{2} be a coset of W_{2}. If C_{1} = C_{2}, then W_{1} = W_{2}.

(66) Let V be a real unitary space, W be a subspace of V, C be a coset of W, and v be a vector of V. Then {v} is a coset of 0_{V} .

(67) Let V be a real unitary space, W be a subspace of V, V_{1} be a subset of the carrier of V, and v be a vector of V. If V_{1} is a coset of 0_{V} , then there exists a vector v of V such that V_{1} = {v}.

(68) For every real unitary space V and for every subspace W of V holds the carrier of W is a coset of W.

(69) For every real unitary space V holds the carrier of V is a coset of Ω_{V}.

(70) Let V be a real unitary space, W be a subspace of V, and V_{1} be a subset of the carrier of V. If V_{1} is a coset of Ω_{V}, then V_{1} = the carrier of V.

(71) Let V be a real unitary space, W be a subspace of V, and C be a coset of W. Then 0_{V} ∈ C if and only if C = the carrier of W.

(72) Let V be a real unitary space, W be a subspace of V, C be a coset of W, and u be a vector of V. Then u ∈ C if and only if C = u + W.

(73) Let V be a real unitary space, W be a subspace of V, C be a coset of W, and u, v be vectors of V. If u ∈ C and v ∈ C, then there exists a vector v_{1} of V such that v_{1} ∈ W and u + v_{1} = v.

(74) Let V be a real unitary space, W be a subspace of V, C be a coset of W,

subspaces and cosets of subspace of real … 7

and u, v be vectors of V. If u ∈ C and v ∈ C, then there exists a vector v_{1} of V such that v_{1} ∈ W and u − v_{1} = v.

(75) Let V be a real unitary space, W be a subspace of V, and v_{1}, v_{2} be vectors of V. Then there exists a coset C of W such that v_{1} ∈ C and v_{2} ∈ C if and only if v_{1} − v_{2} ∈ W.

(76) Let V be a real unitary space, W be a subspace of V, u be a vector of V, and B, C be cosets of W. If u ∈ B and u ∈ C, then B = C.

References

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[3] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[4] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[5] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[6] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[7] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[8] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

[9] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[10] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[11] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

Operations on Subspaces in Real Unitary Space

Noboru Endou Gifu National College of Technology Takashi Mitsuishi Miyagi University

Yasunari Shidama Shinshu University Nagano

Summary. In this article, we extend an operation of real linear space to real unitary space. We show theorems proved in [8] on real unitary space.

MML Identifier: RUSUB 2.

The terminology and notation used here are introduced in the following articles: [7], [3], [10], [11], [2], [1], [13], [12], [6], [9], [5], and [4].

1. Definitions of Sum and Intersection of Subspaces

Let V be a real unitary space and let W_{1}, W_{2} be subspaces of V. The functor W_{1} + W_{2} yields a strict subspace of V and is defined as follows:

(Def. 1) The carrier of W_{1} + W_{2} = {v + u; v ranges over vectors of V, u ranges

over vectors of V: v ∈ W_{1} ∧ u ∈ W_{2}}. Let V be a real unitary space and let W_{1}, W_{2} be subspaces of V. The functor

W_{1} ∩ W_{2} yields a strict subspace of V and is defined by:

(Def. 2) The carrier of W_{1} ∩ W_{2} = (the carrier of W_{1}) ∩ (the carrier of W_{2}).

2. Theorems of Sum and Intersecton of Subspaces

One can prove the following propositions:

(1) Let V be a real unitary space, W_{1}, W_{2} be subspaces of V, and x be a set. Then x ∈ W_{1} + W_{2} if and only if there exist vectors v_{1}, v_{2} of V such that v_{1} ∈ W_{1} and v_{2} ∈ W_{2} and x = v_{1} + v_{2}.

(2) Let V be a real unitary space, W_{1}, W_{2} be subspaces of V, and v be a vector of V. If v ∈ W_{1} or v ∈ W_{2}, then v ∈ W_{1} + W_{2}.

(3) Let V be a real unitary space, W_{1}, W_{2} be subspaces of V, and x be a set. Then x ∈ W_{1} ∩ W_{2} if and only if x ∈ W_{1} and x ∈ W_{2}.

(4) For every real unitary space V and for every strict subspace W of V holds W + W = W.

(5) For every real unitary space V and for all subspaces W_{1}, W_{2} of V holds W_{1} + W_{2} = W_{2} + W_{1}.

(6) For every real unitary space V and for all subspaces W_{1}, W_{2}, W_{3} of V holds W_{1} + (W_{2} + W_{3}) = (W_{1} + W_{2}) + W_{3}.

(7) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Then W_{1} is a subspace of W_{1} + W_{2} and W_{2} is a subspace of W_{1} + W_{2}.

(8) Let V be a real unitary space, W_{1} be a subspace of V, and W_{2} be a strict subspace of V. Then W_{1} is a subspace of W_{2} if and only if W_{1} +W_{2} = W_{2}.

(9) For every real unitary space V and for every strict subspace W of V holds 0_{V} + W = W and W + 0_{V} = W.

(10) Let V be a real unitary space. Then 0_{V} + Ω_{V} = the unitary space structure of V and Ω_{V} + 0_{V} = the unitary space structure of V.

(11) Let V be a real unitary space and W be a subspace of V. Then Ω_{V} +W = the unitary space structure of V and W +Ω_{V} = the unitary space structure of V.

(12) For every strict real unitary space V holds Ω_{V} + Ω_{V} = V.

(13) For every real unitary space V and for every strict subspace W of V holds W ∩ W = W.

(14) For every real unitary space V and for all subspaces W_{1}, W_{2} of V holds W_{1} ∩ W_{2} = W_{2} ∩ W_{1}.

(15) For every real unitary space V and for all subspaces W_{1}, W_{2}, W_{3} of V holds W_{1} ∩ (W_{2} ∩ W_{3}) = (W_{1} ∩ W_{2}) ∩ W_{3}.

(16) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Then W_{1} ∩ W_{2} is a subspace of W_{1} and W_{1} ∩ W_{2} is a subspace of W_{2}.

(17) Let V be a real unitary space, W_{2} be a subspace of V, and W_{1} be a strict subspace of V. Then W_{1} is a subspace of W_{2} if and only if W_{1} ∩ W_{2} = W_{1}.

(18) For every real unitary space V and for every subspace W of V holds 0_{V} ∩ W = 0_{V} and W ∩ 0_{V} = 0_{V} .

(19) For every real unitary space V holds 0_{V} ∩ Ω_{V} = 0_{V} and Ω_{V} ∩ 0_{V} = 0_{V} .

(20) For every real unitary space V and for every strict subspace W of V holds Ω_{V} ∩ W = W and W ∩ Ω_{V} = W.

(21) For every strict real unitary space V holds Ω_{V} ∩ Ω_{V} = V.

(22) For every real unitary space V and for all subspaces W_{1}, W_{2} of V holds W_{1} ∩ W_{2} is a subspace of W_{1} + W_{2}.

(23) For every real unitary space V and for every subspace W_{1} of V and for every strict subspace W_{2} of V holds W_{1} ∩ W_{2} + W_{2} = W_{2}.

(24) For every real unitary space V and for every subspace W_{1} of V and for every strict subspace W_{2} of V holds W_{2} ∩ (W_{2} + W_{1}) = W_{2}.

(25) For every real unitary space V and for all subspaces W_{1}, W_{2}, W_{3} of V holds W_{1} ∩ W_{2} + W_{2} ∩ W_{3} is a subspace of W_{2} ∩ (W_{1} + W_{3}).

(26) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be subspaces of V. If W_{1} is a subspace of W_{2}, then W_{2} ∩ (W_{1} + W_{3}) = W_{1} ∩ W_{2} + W_{2} ∩ W_{3}.

(27) For every real unitary space V and for all subspaces W_{1}, W_{2}, W_{3} of V holds W_{2} + W_{1} ∩ W_{3} is a subspace of (W_{1} + W_{2}) ∩ (W_{2} + W_{3}).

(28) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be subspaces of V. If W_{1} is a subspace of W_{2}, then W_{2} + W_{1} ∩ W_{3} = (W_{1} + W_{2}) ∩ (W_{2} + W_{3}).

(29) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be subspaces of V. If W_{1} is a strict subspace of W_{3}, then W_{1} + W_{2} ∩ W_{3} = (W_{1} + W_{2}) ∩ W_{3}.

(30) For every real unitary space V and for all strict subspaces W_{1}, W_{2} of V holds W_{1} + W_{2} = W_{2} iff W_{1} ∩ W_{2} = W_{1}.

(31) Let V be a real unitary space, W_{1} be a subspace of V, and W_{2}, W_{3} be strict subspaces of V. If W_{1} is a subspace of W_{2}, then W_{1} + W_{3} is a subspace of W_{2} + W_{3}.

(32) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Then there exists a subspace W of V such that the carrier of W = (the carrier of W_{1}) ∪ (the carrier of W_{2}) if and only if W_{1} is a subspace of W_{2} or W_{2} is a subspace of W_{1}.

3. Introduction of a Set of Subspaces of Real Unitary Space

Let V be a real unitary space. The functor SubspacesV yielding a set is defined as follows:

(Def. 3) For every set x holds x ∈ SubspacesV iff x is a strict subspace of V. Let V be a real unitary space. Observe that SubspacesV is non empty. The following proposition is true (33) For every strict real unitary space V holds V ∈ SubspacesV.

4. Definition of the Direct Sum and Linear Complement of Subspaces

Let V be a real unitary space and let W_{1}, W_{2} be subspaces of V. We say that V is the direct sum of W_{1} and W_{2} if and only if:

(Def. 4) The unitary space structure of V = W_{1} + W_{2} and W_{1} ∩ W_{2} = 0_{V} .

Let V be a real unitary space and let W be a subspace of V. A subspace of V is called a linear complement of W if:

(Def. 5) V is the direct sum of it and W.

Let V be a real unitary space and let W be a subspace of V. Observe that there exists a linear complement of W which is strict.

Next we state two propositions:

(34) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Suppose V is the direct sum of W_{1} and W_{2}. Then W_{2} is a linear complement of W_{1}.

(35) Let V be a real unitary space, W be a subspace of V, and L be a linear complement of W. Then V is the direct sum of L and W and the direct sum of W and L.

5. Theorems Concerning the Sum, Linear Complement and Coset of Subspace

The following propositions are true:

(36) Let V be a real unitary space, W be a subspace of V, and L be a linear complement of W. Then W + L = the unitary space structure of V and L + W = the unitary space structure of V.

(37) Let V be a real unitary space, W be a subspace of V, and L be a linear complement of W. Then W ∩ L = 0_{V} and L ∩ W = 0_{V} .

(38) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. If V is the direct sum of W_{1} and W_{2}, then V is the direct sum of W_{2} and W_{1}.

(39) Every real unitary space V is the direct sum of 0_{V} and Ω_{V} and the direct sum of Ω_{V} and 0_{V} .

(40) Let V be a real unitary space, W be a subspace of V, and L be a linear complement of W. Then W is a linear complement of L.

(41) For every real unitary space V holds 0_{V} is a linear complement of Ω_{V} and Ω_{V} is a linear complement of 0_{V} .

(42) Let V be a real unitary space, W_{1}, W_{2} be subspaces of V, C_{1} be a coset of W_{1}, and C_{2} be a coset of W_{2}. If C_{1} meets C_{2}, then C_{1} ∩ C_{2} is a coset of W_{1} ∩ W_{2}.

(43) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Then V is the direct sum of W_{1} and W_{2} if and only if for every coset C_{1} of W_{1} and for every coset C_{2} of W_{2} there exists a vector v of V such that C_{1} ∩ C_{2} = {v}.

6. Decomposition of a Vector of Real Unitary Space

Next we state three propositions:

(44) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Then W_{1} + W_{2} = the unitary space structure of V if and only if for every vector v of V there exist vectors v_{1}, v_{2} of V such that v_{1} ∈ W_{1} and v_{2} ∈ W_{2} and v = v_{1} + v_{2}.

(45) Let V be a real unitary space, W_{1}, W_{2} be subspaces of V, and v, v_{1}, v_{2}, u_{1}, u_{2} be vectors of V. Suppose V is the direct sum of W_{1} and W_{2} and v = v_{1} + v_{2} and v = u_{1} + u_{2} and v_{1} ∈ W_{1} and u_{1} ∈ W_{1} and v_{2} ∈ W_{2} and u_{2} ∈ W_{2}. Then v_{1} = u_{1} and v_{2} = u_{2}.

(46) Let V be a real unitary space and W_{1}, W_{2} be subspaces of V. Suppose

that

(i) V = W_{1} + W_{2}, and (ii) there exists a vector v of V such that for all vectors v_{1}, v_{2}, u_{1}, u_{2} of

V such that v = v_{1} + v_{2} and v = u_{1} + u_{2} and v_{1} ∈ W_{1} and u_{1} ∈ W_{1} and v_{2} ∈ W_{2} and u_{2} ∈ W_{2} holds v_{1} = u_{1} and v_{2} = u_{2}. Then V is the direct sum of W_{1} and W_{2}.

Let V be a real unitary space, let v be a vector of V, and let W_{1}, W_{2} be subspaces of V. Let us assume that V is the direct sum of W_{1} and W_{2}. The functor v_{⟨⟨W_{1},W_{2}⟩⟩} yielding an element of [: the carrier of V, the carrier of V :] is defined as follows:

(Def. 6) v = (v_{⟨⟨W_{1},W_{2}⟩⟩})_{1} + (v_{⟨⟨W_{1},W_{2}⟩⟩})_{2} and (v_{⟨⟨W_{1},W_{2}⟩⟩})_{1} ∈ W_{1} and (v_{⟨⟨W_{1},W_{2}⟩⟩})_{2} ∈

W_{2}. We now state several propositions:

(47) Let V be a real unitary space, v be a vector of V, and W_{1}, W_{2} be subspaces of V. If V is the direct sum of W_{1} and W_{2}, then (v_{⟨⟨W_{1},W_{2}⟩⟩})_{1} = (v_{⟨⟨W_{2},W_{1}⟩⟩})_{2}.

(48) Let V be a real unitary space, v be a vector of V, and W_{1}, W_{2} be subspaces of V. If V is the direct sum of W_{1} and W_{2}, then (v_{⟨⟨W_{1},W_{2}⟩⟩})_{2} = (v_{⟨⟨W_{2},W_{1}⟩⟩})_{1}.

(49) Let V be a real unitary space, W be a subspace of V, L be a linear complement of W, v be a vector of V, and t be an element of [: the carrier of V, the carrier of V :]. If t_{1} + t_{2} = v and t_{1} ∈ W and t_{2} ∈ L, then t = v_{⟨⟨W,L⟩⟩}.

(50) Let V be a real unitary space, W be a subspace of V, L be a linear complement of W, and v be a vector of V. Then (v_{⟨⟨W,L⟩⟩})_{1}+(v_{⟨⟨W,L⟩⟩})_{2} = v.

(51) Let V be a real unitary space, W be a subspace of V, L be a linear complement of W, and v be a vector of V. Then (v_{⟨⟨W,L⟩⟩})_{1} ∈ W and (v_{⟨⟨W,L⟩⟩})_{2} ∈ L.

(52) Let V be a real unitary space, W be a subspace of V, L be a linear complement of W, and v be a vector of V. Then (v_{⟨⟨W,L⟩⟩})_{1} = (v_{⟨⟨L,W ⟩⟩})_{2}.

(53) Let V be a real unitary space, W be a subspace of V, L be a linear complement of W, and v be a vector of V. Then (v_{⟨⟨W,L⟩⟩})_{2} = (v_{⟨⟨L,W ⟩⟩})_{1}.

7. Introduction of Operations on Set of Subspaces

Let V be a real unitary space. The functor SubJoinV yields a binary operation on SubspacesV and is defined by:

(Def. 7) For all elements A_{1}, A_{2} of SubspacesV and for all subspaces W_{1}, W_{2} of V

such that A_{1} = W_{1} and A_{2} = W_{2} holds (SubJoinV)(A_{1}, A_{2}) = W_{1} + W_{2}. Let V be a real unitary space. The functor SubMeetV yielding a binary

operation on SubspacesV is defined as follows: (Def. 8) For all elements A_{1}, A_{2} of SubspacesV and for all subspaces W_{1}, W_{2} of V

such that A_{1} = W_{1} and A_{2} = W_{2} holds (SubMeetV)(A_{1}, A_{2}) = W_{1} ∩ W_{2}.

8. Theorems of Functions SubJoin, SubMeet

We now state the proposition

(54) For every real unitary space V holds ⟨SubspacesV, SubJoinV,

SubMeetV ⟩ is a lattice. Let V be a real unitary space. Note that ⟨SubspacesV, SubJoinV, SubMeetV ⟩

is lattice-like.

The following propositions are true:

(55) For every real unitary space V holds ⟨SubspacesV, SubJoinV, SubMeetV ⟩ is lower-bounded.

(56) For every real unitary space V holds ⟨SubspacesV, SubJoinV, SubMeetV ⟩ is upper-bounded.

operations on subspaces in real unitary space 15

(57) For every real unitary space V holds ⟨SubspacesV, SubJoinV, SubMeetV ⟩ is a bound lattice.

(58) For every real unitary space V holds ⟨SubspacesV, SubJoinV, SubMeetV ⟩ is modular.

(59) For every real unitary space V holds ⟨SubspacesV, SubJoinV,

SubMeetV ⟩ is complemented. Let V be a real unitary space. Observe that ⟨SubspacesV, SubJoinV, SubMeetV ⟩ is lower-bounded, upper-

bounded, modular, and complemented.

One can prove the following proposition

(60) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be strict subspaces of V. If W_{1} is a subspace of W_{2}, then W_{1} ∩ W_{3} is a subspace of W_{2} ∩ W_{3}.

9. Auxiliary Theorems in Real Unitary Space

We now state three propositions:

(61) Let V be a real unitary space and W be a strict subspace of V. Suppose that for every vector v of V holds v ∈ W. Then W = the unitary space structure of V.

(62) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. Then there exists a coset C of W such that v ∈ C.

(63) Let V be a real unitary space, W be a subspace of V, v be a vector of V, and x be a set. Then x ∈ v + W if and only if there exists a vector u of V such that u ∈ W and x = v + u.

References

[1] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[3] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[4] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Subspaces and cosets of subspace of real unitary space. Formalized Mathematics, 11(1):1–7, 2003.

[5] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[6] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[7] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[8] Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics, 1(2):395–399, 1990.

[9] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[10] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[11] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

16 noboru endou et al.

[12] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[13] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

Linear Combinations in Real Unitary Space

Noboru Endou Gifu National College of Technology Takashi Mitsuishi Miyagi University

Yasunari Shidama Shinshu University Nagano

Summary. In this article, we mainly discuss linear combination of vectors in Real Unitary Space and dimension of the space. As the result, we obtain some theorems that are similar to those in Real Linear Space.

MML Identifier: RUSUB 3.

The articles [11], [5], [16], [2], [17], [1], [3], [4], [15], [10], [6], [14], [13], [9], [12], [8], and [7] provide the terminology and notation for this paper.

1. Definition and Fundamental Properties of Linear Combination

Let V be a real unitary space and let A be a subset of the carrier of V. The functor Lin(A) yielding a strict subspace of V is defined by:

(Def. 1) The carrier of Lin(A) = {∑ l : l ranges over linear combinations of A}. We now state a number of propositions:

(1) Let V be a real unitary space, A be a subset of the carrier of V, and x be a set. Then x ∈ Lin(A) if and only if there exists a linear combination l of A such that x = ∑ l.

(2) Let V be a real unitary space, A be a subset of the carrier of V, and x be a set. If x ∈ A, then x ∈ Lin(A).

(3) For every real unitary space V holds Lin(∅_{the carrier of V}) = 0_{V} .

(4) For every real unitary space V and for every subset A of the carrier of V such that Lin(A) = 0_{V} holds A = ∅ or A = {0_{V} }.

(5) Let V be a real unitary space, W be a strict subspace of V, and A be a subset of the carrier of V. If A = the carrier of W, then Lin(A) = W.

(6) Let V be a strict real unitary space and A be a subset of the carrier of V. If A = the carrier of V, then Lin(A) = V.

(7) Let V be a real unitary space and A, B be subsets of the carrier of V. If A ⊆ B, then Lin(A) is a subspace of Lin(B).

(8) Let V be a strict real unitary space and A, B be subsets of the carrier of V. If Lin(A) = V and A ⊆ B, then Lin(B) = V.

(9) For every real unitary space V and for all subsets A, B of the carrier of V holds Lin(A ∪ B) = Lin(A) + Lin(B).

(10) For every real unitary space V and for all subsets A, B of the carrier of V holds Lin(A ∩ B) is a subspace of Lin(A) ∩ Lin(B).

(11) Let V be a real unitary space and A be a subset of the carrier of V. Suppose A is linearly independent. Then there exists a subset B of the carrier of V such that A ⊆ B and B is linearly independent and Lin(B) = the unitary space structure of V.

(12) Let V be a real unitary space and A be a subset of the carrier of V. Suppose Lin(A) = V. Then there exists a subset B of the carrier of V such that B ⊆ A and B is linearly independent and Lin(B) = V.

2. Definition of the Basis of Real Unitary Space

Let V be a real unitary space. A subset of the carrier of V is said to be a basis of V if:

(Def. 2) It is linearly independent and Lin(it) = the unitary space structure of

V. One can prove the following three propositions:

(13) Let V be a strict real unitary space and A be a subset of the carrier of V. If A is linearly independent, then there exists a basis I of V such that A ⊆ I.

(14) Let V be a real unitary space and A be a subset of the carrier of V. If Lin(A) = V, then there exists a basis I of V such that I ⊆ A.

(15) Let V be a real unitary space and A be a subset of V. If A is linearly independent, then there exists a basis I of V such that A ⊆ I.

3. Some Theorems of Lin, Sum, Carrier

We now state a number of propositions:

(16) Let V be a real unitary space, L be a linear combination of V, A be a subset of V, and F be a finite sequence of elements of the carrier of V. Suppose rngF ⊆ the carrier of Lin(A). Then there exists a linear combination K of A such that ∑(L F) = ∑ K.

(17) Let V be a real unitary space, L be a linear combination of V, and A be a subset of V. Suppose the support of L ⊆ the carrier of Lin(A). Then there exists a linear combination K of A such that ∑ L = ∑ K.

(18) Let V be a real unitary space, W be a subspace of V, and L be a linear combination of V. Suppose the support of L ⊆ the carrier of W. Let K be a linear combination of W. Suppose K = L↾the carrier of W. Then the support of L = the support of K and ∑ L = ∑ K.

(19) Let V be a real unitary space, W be a subspace of V, and K be a linear combination of W. Then there exists a linear combination L of V such that the support of K = the support of L and ∑ K = ∑ L.

(20) Let V be a real unitary space, W be a subspace of V, and L be a linear combination of V. Suppose the support of L ⊆ the carrier of W. Then there exists a linear combination K of W such that the support of K = the support of L and ∑ K = ∑ L.

(21) For every real unitary space V and for every basis I of V and for every vector v of V holds v ∈ Lin(I).

(22) Let V be a real unitary space, W be a subspace of V, and A be a subset of W. Suppose A is linearly independent. Then there exists a subset B of V such that B is linearly independent and B = A.

(23) Let V be a real unitary space, W be a subspace of V, and A be a subset of V. Suppose A is linearly independent and A ⊆ the carrier of W. Then there exists a subset B of W such that B is linearly independent and B = A.

(24) Let V be a real unitary space, W be a subspace of V, and A be a basis of W. Then there exists a basis B of V such that A ⊆ B.

(25) Let V be a real unitary space and A be a subset of V. Suppose A is linearly independent. Let v be a vector of V. If v ∈ A, then for every subset B of V such that B = A \ {v} holds v ∉ Lin(B).

(26) Let V be a real unitary space, I be a basis of V, and A be a non empty subset of V. Suppose A misses I. Let B be a subset of V. If B = I ∪ A, then B is linearly dependent.

(27) Let V be a real unitary space, W be a subspace of V, and A be a subset of V. If A ⊆ the carrier of W, then Lin(A) is a subspace of W.

(28) Let V be a real unitary space, W be a subspace of V, A be a subset of V, and B be a subset of W. If A = B, then Lin(A) = Lin(B).

4. Subspaces of Real Unitary Space Generated by One, Two, or Three Vectors

We now state a number of propositions:

(29) Let V be a real unitary space, v be a vector of V, and x be a set. Then x ∈ Lin({v}) if and only if there exists a real number a such that x = a·v.

(30) For every real unitary space V and for every vector v of V holds v ∈ Lin({v}).

(31) Let V be a real unitary space, v, w be vectors of V, and x be a set. Then x ∈ v + Lin({w}) if and only if there exists a real number a such that x = v + a · w.

(32) Let V be a real unitary space, w_{1}, w_{2} be vectors of V, and x be a set. Then x ∈ Lin({w_{1}, w_{2}}) if and only if there exist real numbers a, b such that x = a · w_{1} + b · w_{2}.

(33) For every real unitary space V and for all vectors w_{1}, w_{2} of V holds w_{1} ∈ Lin({w_{1}, w_{2}}) and w_{2} ∈ Lin({w_{1}, w_{2}}).

(34) Let V be a real unitary space, v, w_{1}, w_{2} be vectors of V, and x be a set. Then x ∈ v + Lin({w_{1}, w_{2}}) if and only if there exist real numbers a, b such that x = v + a · w_{1} + b · w_{2}.

(35) Let V be a real unitary space, v_{1}, v_{2}, v_{3} be vectors of V, and x be a set. Then x ∈ Lin({v_{1}, v_{2}, v_{3}}) if and only if there exist real numbers a, b, c such that x = a · v_{1} + b · v_{2} + c · v_{3}.

(36) For every real unitary space V and for all vectors w_{1}, w_{2}, w_{3} of V holds w_{1} ∈ Lin({w_{1}, w_{2}, w_{3}}) and w_{2} ∈ Lin({w_{1}, w_{2}, w_{3}}) and w_{3} ∈ Lin({w_{1}, w_{2}, w_{3}}).

(37) Let V be a real unitary space, v, w_{1}, w_{2}, w_{3} be vectors of V, and x be a set. Then x ∈ v + Lin({w_{1}, w_{2}, w_{3}}) if and only if there exist real numbers a, b, c such that x = v + a · w_{1} + b · w_{2} + c · w_{3}.

(38) For every real unitary space V and for all vectors v, w of V such that v ∈ Lin({w}) and v ≠ 0_{V} holds Lin({v}) = Lin({w}).

(39) Let V be a real unitary space and v_{1}, v_{2}, w_{1}, w_{2} be vectors of V. Suppose v_{1} ≠ v_{2} and {v_{1}, v_{2}} is linearly independent and v_{1} ∈ Lin({w_{1}, w_{2}}) and v_{2} ∈ Lin({w_{1}, w_{2}}). Then Lin({w_{1}, w_{2}}) = Lin({v_{1}, v_{2}}) and {w_{1}, w_{2}} is linearly independent and w_{1} ≠ w_{2}. linear combinations in real unitary space 21

5. Auxiliary Theorems

We now state several propositions:

(40) For every real unitary space V and for every set x holds x ∈ 0_{V} iff x = 0_{V} .

(41) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be subspaces of V. If W_{1} is a subspace of W_{3}, then W_{1} ∩ W_{2} is a subspace of W_{3}.

(42) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be subspaces of V. Suppose W_{1} is a subspace of W_{2} and a subspace of W_{3}. Then W_{1} is a subspace of W_{2} ∩ W_{3}.

(43) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be subspaces of V. Suppose W_{1} is a subspace of W_{3} and W_{2} is a subspace of W_{3}. Then W_{1}+W_{2} is a subspace of W_{3}.

(44) Let V be a real unitary space and W_{1}, W_{2}, W_{3} be subspaces of V. If W_{1} is a subspace of W_{2}, then W_{1} is a subspace of W_{2} + W_{3}.

(45) Let V be a real unitary space, W_{1}, W_{2} be subspaces of V, and v be a vector of V. If W_{1} is a subspace of W_{2}, then v + W_{1} ⊆ v + W_{2}.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[7] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Operations on subspaces in real unitary space. Formalized Mathematics, 11(1):9–16, 2003.

[8] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Subspaces and cosets of subspace of real unitary space. Formalized Mathematics, 11(1):1–7, 2003.

[9] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[10] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[12] Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847–850, 1990.

[13] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.

[14] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

[15] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[16] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[17] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Dimension of Real Unitary Space

Noboru Endou

Gifu National College of Technology Takashi Mitsuishi Miyagi University

Yasunari Shidama Shinshu University Nagano

Summary. In this article we describe the dimension of real unitary space. Most of theorems are restricted from real linear space. In the last section, we introduce affine subset of real unitary space.

MML Identifier: RUSUB 4.

The papers [14], [13], [19], [2], [3], [4], [1], [5], [11], [18], [6], [10], [17], [16], [12], [15], [9], [8], and [7] provide the terminology and notation for this paper.

1. Finite-dimensional Real Unitary Space

One can prove the following two propositions:

(1) Let V be a real unitary space, A, B be finite subsets of V, and v be a vector of V. Suppose v ∈ Lin(A ∪ B) and v ∉ Lin(B). Then there exists a vector w of V such that w ∈ A and w ∈ Lin(((A ∪ B) \ {w}) ∪ {v}).

(2) Let V be a real unitary space and A, B be finite subsets of V. Suppose the unitary space structure of V = Lin(A) and B is linearly independent. Then \overline{\overline{B}} \overline{\overline{A}} and there exists a finite subset C of V such that C ⊆ A and \overline{\overline{C}} = \overline{\overline{A}} − \overline{\overline{B}} and the unitary space structure of V = Lin(B ∪ C).

Let V be a real unitary space. We say that V is finite dimensional if and only if:

(Def. 1) There exists a finite subset of the carrier of V which is a basis of V.

Let us mention that there exists a real unitary space which is strict and finite dimensional.

Let V be a real unitary space. Let us observe that V is finite dimensional if and only if:

(Def. 2) There exists a finite subset of V which is a basis of V. We now state several propositions:

(3) For every real unitary space V such that V is finite dimensional holds every basis of V is finite.

(4) Let V be a real unitary space and A be a subset of V. Suppose V is finite dimensional and A is linearly independent. Then A is finite.

(5) For every real unitary space V and for all bases A, B of V such that V is finite dimensional holds \overline{\overline{A}} = \overline{\overline{B}} .

(6) For every real unitary space V holds 0_{V} is finite dimensional.

(7) Let V be a real unitary space and W be a subspace of V. If V is finite dimensional, then W is finite dimensional.

Let V be a real unitary space. Note that there exists a subspace of V which is finite dimensional and strict.

Let V be a finite dimensional real unitary space. Observe that every subspace of V is finite dimensional.

Let V be a finite dimensional real unitary space. Observe that there exists a subspace of V which is strict.

2. Dimension of Real Unitary Space

Let V be a real unitary space. Let us assume that V is finite dimensional. The functor dim(V) yielding a natural number is defined by:

(Def. 3) For every basis I of V holds dim(V) = \overline{\overline{I}} .

One can prove the following propositions:

(8) For every finite dimensional real unitary space V and for every subspace W of V holds dim(W) dim(V).

(9) Let V be a finite dimensional real unitary space and A be a subset of V. If A is linearly independent, then \overline{\overline{A}} = dim(Lin(A)).

(10) For every finite dimensional real unitary space V holds dim(V) = dim(Ω_{V}).

(11) Let V be a finite dimensional real unitary space and W be a subspace of V. Then dim(V) = dim(W) if and only if Ω_{V} = Ω_{W} .

(12) For every finite dimensional real unitary space V holds dim(V) = 0 iff Ω_{V} = 0_{V} .

(13) Let V be a finite dimensional real unitary space. Then dim(V) = 1 if and only if there exists a vector v of V such that v ≠ 0_{V} and Ω_{V} = Lin({v}).

(14) Let V be a finite dimensional real unitary space. Then dim(V) = 2 if and only if there exist vectors u, v of V such that u ≠ v and {u, v} is linearly independent and Ω_{V} = Lin({u, v}).

(15) For every finite dimensional real unitary space V and for all subspaces W_{1}, W_{2} of V holds dim(W_{1} +W_{2})+dim(W_{1} ∩W_{2}) = dim(W_{1})+dim(W_{2}).

(16) For every finite dimensional real unitary space V and for all subspaces W_{1}, W_{2} of V holds dim(W_{1} ∩ W_{2}) (dim(W_{1}) + dim(W_{2})) − dim(V).

(17) Let V be a finite dimensional real unitary space and W_{1}, W_{2} be subspaces of V. If V is the direct sum of W_{1} and W_{2}, then dim(V) = dim(W_{1}) + dim(W_{2}).

3. Fixed-dimensional Subspace Family

We now state the proposition

(18) Let V be a finite dimensional real unitary space, W be a subspace of V, and n be a natural number. Then n dim(V) if and only if there exists a strict subspace W of V such that dim(W) = n.

Let V be a finite dimensional real unitary space and let n be a natural number. The functor Sub_{n}(V) yields a set and is defined as follows:

(Def. 4) For every set x holds x ∈ Sub_{n}(V) iff there exists a strict subspace W

of V such that W = x and dim(W) = n. Next we state three propositions:

(19) Let V be a finite dimensional real unitary space and n be a natural number. If n dim(V), then Sub_{n}(V) is non empty.

(20) For every finite dimensional real unitary space V and for every natural number n such that dim(V) < n holds Sub_{n}(V) = ∅.

(21) Let V be a finite dimensional real unitary space, W be a subspace of V, and n be a natural number. Then Sub_{n}(W) ⊆ Sub_{n}(V).

4. Affine Set

Let V be a non empty RLS structure and let S be a subset of V. We say that S is Affine if and only if:

(Def. 5) For all vectors x, y of V and for every real number a such that x ∈ S

and y ∈ S holds (1 − a) · x + a · y ∈ S. One can prove the following propositions:

(22) For every non empty RLS structure V holds Ω_{V} is Affine and ∅_{V} is Affine.

(23) For every real linear space-like non empty RLS structure V and for every

vector v of V holds {v} is Affine. Let V be a non empty RLS structure. Observe that there exists a subset of

V which is non empty and Affine and there exists a subset of V which is empty and Affine.

Let V be a real linear space and let W be a subspace of V. The functor Up(W) yielding a non empty subset of V is defined by:

(Def. 6) Up(W) = the carrier of W.

Let V be a real unitary space and let W be a subspace of V. The functor Up(W) yielding a non empty subset of V is defined by:

(Def. 7) Up(W) = the carrier of W.

We now state two propositions:

(24) For every real linear space V and for every subspace W of V holds Up(W) is Affine and 0_{V} ∈ the carrier of W.

(25) Let V be a real linear space and A be a Affine subset of V. Suppose 0_{V} ∈ A. Let x be a vector of V and a be a real number. If x ∈ A, then a · x ∈ A.

Let V be a non empty RLS structure and let S be a non empty subset of V. We say that S is Subspace-like if and only if the conditions (Def. 8) are satisfied.

(Def. 8)(i) The zero of V ∈ S, and

(ii) for all elements x, y of the carrier of V and for every real number a

such that x ∈ S and y ∈ S holds x + y ∈ S and a · x ∈ S. One can prove the following propositions:

(26) Let V be a real linear space and A be a non empty Affine subset of V. If 0_{V} ∈ A, then A is Subspace-like and A = the carrier of Lin(A).

(27) For every real linear space V and for every subspace W of V holds Up(W) is Subspace-like.

(28) For every real linear space V and for every strict subspace W of V holds W = Lin(Up(W)).

(29) Let V be a real unitary space and A be a non empty Affine subset of V. If 0_{V} ∈ A, then A = the carrier of Lin(A).

(30) For every real unitary space V and for every subspace W of V holds Up(W) is Subspace-like.

(31) For every real unitary space V and for every strict subspace W of V

holds W = Lin(Up(W)). Let V be a non empty loop structure, let M be a subset of the carrier of V,

and let v be an element of the carrier of V. The functor v + M yields a subset

of V and is defined as follows: (Def. 9) v + M = {v + u; u ranges over elements of the carrier of V: u ∈ M}.

We now state three propositions:

(32) Let V be a real linear space, W be a strict subspace of V, M be a subset of the carrier of V, and v be a vector of V. If Up(W) = M, then v + W = v + M.

(33) Let V be an Abelian add-associative real linear space-like non empty RLS structure, M be a Affine subset of V, and v be a vector of V. Then v + M is Affine.

(34) Let V be a real unitary space, W be a strict subspace of V, M be a subset of the carrier of V, and v be a vector of V. If Up(W) = M, then v + W = v + M.

Let V be a non empty loop structure and let M, N be subsets of the carrier of V. The functor M + N yields a subset of V and is defined as follows:

(Def. 10) M + N = {u + v; u ranges over elements of the carrier of V, v ranges

over elements of the carrier of V: u ∈ M ∧ v ∈ N}. We now state the proposition

(35) For every Abelian non empty loop structure V and for all subsets M, N

of V holds N + M = M + N. Let V be an Abelian non empty loop structure and let M, N be subsets of

V. Let us observe that the functor M + N is commutative.

Next we state four propositions:

(36) Let V be a non empty loop structure, M be a subset of V, and v be an element of the carrier of V. Then {v} + M = v + M.

(37) Let V be an Abelian add-associative real linear space-like non empty RLS structure, M be a Affine subset of V, and v be a vector of V. Then {v} + M is Affine.

(38) For every non empty RLS structure V and for all Affine subsets M, N of V holds M ∩ N is Affine.

(39) Let V be an Abelian add-associative real linear space-like non empty RLS structure and M, N be Affine subsets of V. Then M + N is Affine.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, [6] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[7] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Linear combinations in real unitary space. Formalized Mathematics, 11(1):17–22, 2003.

[8] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Operations on subspaces in real unitary space. Formalized Mathematics, 11(1):9–16, 2003.

[9] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Subspaces and cosets of subspace of real unitary space. Formalized Mathematics, 11(1):1–7, 2003.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[11] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[12] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[13] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[14] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[15] Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847–850, 1990.

[16] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.

[17] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

[18] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

Trigonometric Functions on Complex Space

Takashi Mitsuishi Miyagi University

Noboru Endou Gifu National College of Technology

Keiji Ohkubo Shinshu University Nagano

Summary. This article describes definitions of sine, cosine, hyperbolic sine and hyperbolic cosine. Some of their basic properties are discussed.

MML Identifier: SIN COS3.

The notation and terminology used here are introduced in the following papers: [9], [4], [10], [1], [8], [2], [3], [5], [7], [11], and [6].

1. Definitions of Trigonometric Functions

We adopt the following convention: x, y denote elements of ℝ, z, z_{1}, z_{2} denote elements of ℂ, and n denotes a natural number.

The function sin_{ℂ} from ℂ into ℂ is defined by: (Def. 1) sin_{ℂ}(z) = \frac{exp(i·z)−exp(− i·z)}{(2+0i)·i}.

The function cos_{ℂ} from ℂ into ℂ is defined by: (Def. 2) cos_{ℂ}(z) = \frac{exp(i·z)+exp(− i·z)}{2+0i}.

The function sinh_{ℂ} from ℂ into ℂ is defined by: (Def. 3) sinh_{ℂ}(z) = \frac{exp z−exp(− z)}{2+0i} .

The function cosh_{ℂ} from ℂ into ℂ is defined by: (Def. 4) cosh_{ℂ}(z) = \frac{exp z+exp(− z)}{2+0i}.

2. Properties of Trigonometric Functions on Complex Space

We now state a number of propositions:

(1) For every element z of ℂ holds sin_{ℂ}_{z} · sin_{ℂ}_{z} + cos_{ℂ}_{z} · cos_{ℂ}_{z} = 1_{ℂ}.

(2) − sin_{ℂ}_{z} = sin_{ℂ}_{− z} .

(3) cos_{ℂ}_{z} = cos_{ℂ}_{− z} .

(4) sin_{ℂ}_{z_{1}+z_{2}} = sin_{ℂ}_{z_{1}} · cos_{ℂ}_{z_{2}} + cos_{ℂ}_{z_{1}} · sin_{ℂ}_{z_{2}} .

(5) sin_{ℂ}_{z_{1}−z_{2}} = sin_{ℂ}_{z_{1}} · cos_{ℂ}_{z_{2}} − cos_{ℂ}_{z_{1}} · sin_{ℂ}_{z_{2}} .

(6) cos_{ℂ}_{z_{1}+z_{2}} = cos_{ℂ}_{z_{1}} · cos_{ℂ}_{z_{2}} − sin_{ℂ}_{z_{1}} · sin_{ℂ}_{z_{2}} .

(7) cos_{ℂ}_{z_{1}−z_{2}} = cos_{ℂ}_{z_{1}} · cos_{ℂ}_{z_{2}} + sin_{ℂ}_{z_{1}} · sin_{ℂ}_{z_{2}} .

(8) cosh_{ℂ}_{z} · cosh_{ℂ}_{z} − sinh_{ℂ}_{z} · sinh_{ℂ}_{z} = 1_{ℂ}.

(9) − sinh_{ℂ}_{z} = sinh_{ℂ}_{− z} .

(10) cosh_{ℂ}_{z} = cosh_{ℂ}_{− z} .

(11) sinh_{ℂ}_{z_{1}+z_{2}} = sinh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{2}} + cosh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{2}} .

(12) sinh_{ℂ}_{z_{1}−z_{2}} = sinh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{2}} − cosh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{2}} .

(13) cosh_{ℂ}_{z_{1}−z_{2}} = cosh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{2}} − sinh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{2}} .

(14) cosh_{ℂ}_{z_{1}+z_{2}} = cosh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{2}} + sinh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{2}} .

(15) sin_{ℂ}_{i·z} = i · sinh_{ℂ}_{z} .

(16) cos_{ℂ}_{i·z} = cosh_{ℂ}_{z} .

(17) sinh_{ℂ}_{i·z} = i · sin_{ℂ}_{z} .

(18) cosh_{ℂ}_{i·z} = cos_{ℂ}_{z} .

(19) For all elements x, y of ℝ holds exp(x + yi) = exp(x) · cos(y) + (exp(x) · sin(y))i.

(20) exp(0_{ℂ}) = 1 + 0i.

(21) sin_{ℂ}_{0_{ℂ}} = 0_{ℂ}.

(22) sinh_{ℂ}_{0_{ℂ}} = 0_{ℂ}.

(23) cos_{ℂ}_{0_{ℂ}} = 1 + 0i.

(24) cosh_{ℂ}_{0_{ℂ}} = 1 + 0i.

(25) expz = cosh_{ℂ}_{z} + sinh_{ℂ}_{z} .

(26) exp(− z) = cosh_{ℂ}_{z} − sinh_{ℂ}_{z} .

(27) exp(z + (2 · π + 0i) · i) = expz and exp(z + (0 + (2 · π)i)) = expz.

(28) exp(0 + (2 · π · n)i) = 1 + 0i and exp((2 · π · n + 0i) · i) = 1 + 0i.

(29) exp(0 + (− 2 · π · n)i) = 1 + 0i and exp((− 2 · π · n+ 0i) · i) = 1 + 0i.

(30) exp(0+((2·n+1)·π)i) = − 1+0i and exp(((2·n+1)·π+0i)·i) = − 1+0i.

(31) exp(0+(− (2 · n + 1) · π)i) = − 1+0i and exp((− (2 · n + 1) · π+0i)·i) = − 1 + 0i.

(32) exp(0 + ((2 · n + \frac{1}{2}) · π)i) = 0 + 1i and exp(((2 · n + \frac{1}{2}) · π + 0i) · i) = 0 + 1i.

(33) exp(0+(− (2 · n + \frac{1}{2}) · π)i) = 0+(− 1)i and exp((− (2 · n + \frac{1}{2}) · π+0i)·i) = 0 + (− 1)i.

(34) sin_{ℂ}_{z+(2·n·π+0i)}= sin_{ℂ}_{z} .

(35) cos_{ℂ}_{z+(2·n·π+0i)}= cos_{ℂ}_{z} .

(36) exp(i · z) = cos_{ℂ}_{z} +i · sin_{ℂ}_{z} .

(37) exp(− i · z) = cos_{ℂ}_{z} −i · sin_{ℂ}_{z} .

(38) For every element x of ℝ holds sin_{ℂ}_{x+0i} = sin(x) + 0i.

(39) For every element x of ℝ holds cos_{ℂ}_{x+0i} = cos(x) + 0i.

(40) For every element x of ℝ holds sinh_{ℂ}_{x+0i} = sinh(x) + 0i.

(41) For every element x of ℝ holds cosh_{ℂ}_{x+0i} = cosh(x) + 0i.

(42) For all elements x, y of ℝ holds x + yi = (x + 0i) + i · (y + 0i).

(43) sin_{ℂ}_{x+yi} = sin(x) · cosh(y) + (cos(x) · sinh(y))i.

(44) sin_{ℂ}_{x+(− y)i} = sin(x) · cosh(y) + (− cos(x) · sinh(y))i.

(45) cos_{ℂ}_{x+yi} = cos(x) · cosh(y) + (− sin(x) · sinh(y))i.

(46) cos_{ℂ}_{x+(− y)i} = cos(x) · cosh(y) + (sin(x) · sinh(y))i.

(47) sinh_{ℂ}_{x+yi} = sinh(x) · cos(y) + (cosh(x) · sin(y))i.

(48) sinh_{ℂ}_{x+(− y)i} = sinh(x) · cos(y) + (− cosh(x) · sin(y))i.

(49) cosh_{ℂ}_{x+yi} = cosh(x) · cos(y) + (sinh(x) · sin(y))i.

(50) cosh_{ℂ}_{x+(− y)i} = cosh(x) · cos(y) + (− sinh(x) · sin(y))i.

(51) For every natural number n and for every element z of ℂ holds (cos_{ℂ}_{z} +i· sin_{ℂ}_{z})_{ℕ}^{n} = cos_{ℂ}_{(n+0i)·z}+i · sin_{ℂ}_{(n+0i)·z}.

(52) For every natural number n and for every element z of ℂ holds (cos_{ℂ}_{z} −i· sin_{ℂ}_{z})_{ℕ}^{n} = cos_{ℂ}_{(n+0i)·z}−i · sin_{ℂ}_{(n+0i)·z}.

(53) For every natural number n and for every element z of ℂ holds exp(i · (n + 0i) · z) = (cos_{ℂ}_{z} +i · sin_{ℂ}_{z})_{ℕ}^{n}.

(54) For every natural number n and for every element z of ℂ holds exp(− i · (n + 0i) · z) = (cos_{ℂ}_{z} −i · sin_{ℂ}_{z})_{ℕ}^{n}.

(55) For all elements x, y of ℝ holds \frac{1+(− 1)i}{2+0i} · sinh_{ℂ}_{x+yi} +\frac{1+1i}{2+0i} · sinh_{ℂ}_{x+(− y)i} = (sinh(x) · cos(y) + cosh(x) · sin(y)) + 0i.

(56) For all elements x, y of ℝ holds \frac{1+(− 1)i}{2+0i} ·cosh_{ℂ}_{x+yi} +\frac{1+1i}{2+0i} ·cosh_{ℂ}_{x+(− y)i} = (sinh(x) · sin(y) + cosh(x) · cos(y)) + 0i.

(57) sinh_{ℂ}_{z} · sinh_{ℂ}_{z} = \frac{cosh_{ℂ}_{(2+0i)·z}−(1+0i)}{2+0i} .

(58) cosh_{ℂ}_{z} · cosh_{ℂ}_{z} = \frac{cosh_{ℂ}_{(2+0i)·z}+(1+0i)}{2+0i}.

(59) sinh_{ℂ}_{(2+0i)·z} = (2 + 0i) · sinh_{ℂ}_{z} · cosh_{ℂ}_{z} and cosh_{ℂ}_{(2+0i)·z} = (2 + 0i) · cosh_{ℂ}_{z} · cosh_{ℂ}_{z} −(1 + 0i).

(60) sinh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{1}} − sinh_{ℂ}_{z_{2}} · sinh_{ℂ}_{z_{2}} = sinh_{ℂ}_{z_{1}+z_{2}} · sinh_{ℂ}_{z_{1}−z_{2}} and cosh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{1}} − cosh_{ℂ}_{z_{2}} · cosh_{ℂ}_{z_{2}} = sinh_{ℂ}_{z_{1}+z_{2}} · sinh_{ℂ}_{z_{1}−z_{2}} and 32 takashi mitsuishi et al.

sinh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{1}} − sinh_{ℂ}_{z_{2}} · sinh_{ℂ}_{z_{2}} = cosh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{1}} − cosh_{ℂ}_{z_{2}} · cosh_{ℂ}_{z_{2}} .

(61) cosh_{ℂ}_{z_{1}+z_{2}} · cosh_{ℂ}_{z_{1}−z_{2}} = sinh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{1}} + cosh_{ℂ}_{z_{2}} · cosh_{ℂ}_{z_{2}} and cosh_{ℂ}_{z_{1}+z_{2}} · cosh_{ℂ}_{z_{1}−z_{2}} = cosh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{1}} + sinh_{ℂ}_{z_{2}} · sinh_{ℂ}_{z_{2}} and sinh_{ℂ}_{z_{1}} · sinh_{ℂ}_{z_{1}} + cosh_{ℂ}_{z_{2}} · cosh_{ℂ}_{z_{2}} = cosh_{ℂ}_{z_{1}} · cosh_{ℂ}_{z_{1}} + sinh_{ℂ}_{z_{2}} · sinh_{ℂ}_{z_{2}} .

(62) sinh_{ℂ}_{(2+0i)·z_{1}}+ sinh_{ℂ}_{(2+0i)·z_{2}} = (2 + 0i) · sinh_{ℂ}_{z_{1}+z_{2}} · cosh_{ℂ}_{z_{1}−z_{2}} and sinh_{ℂ}_{(2+0i)·z_{1}}− sinh_{ℂ}_{(2+0i)·z_{2}} = (2 + 0i) · sinh_{ℂ}_{z_{1}−z_{2}} · cosh_{ℂ}_{z_{1}+z_{2}} .

(63) cosh_{ℂ}_{(2+0i)·z_{1}}+ cosh_{ℂ}_{(2+0i)·z_{2}} = (2 + 0i) · cosh_{ℂ}_{z_{1}+z_{2}} · cosh_{ℂ}_{z_{1}−z_{2}} and cosh_{ℂ}_{(2+0i)·z_{1}}− cosh_{ℂ}_{(2+0i)·z_{2}} = (2 + 0i) · sinh_{ℂ}_{z_{1}+z_{2}} · sinh_{ℂ}_{z_{1}−z_{2}} .

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[5] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[6] Takashi Mitsuishi and Yuguang Yang. Properties of the trigonometric function. Formalized Mathematics, 8(1):103–106, 1999.

[7] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403–410, 1997.

[8] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[9] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[10] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[11] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

Topology of Real Unitary Space

Noboru Endou

Gifu National College of Technology Takashi Mitsuishi Miyagi University

Yasunari Shidama Shinshu University Nagano

Summary. In this article we introduce three subjects in real unitary space: parallelism of subsets, orthogonality of subsets and topology of the space. In particular, to introduce the topology of real unitary space, we discuss the metric topology which is induced by the inner product in the space. As the result, we are able to discuss some topological subjects on real unitary space.

MML Identifier: RUSUB 5.

The articles [8], [12], [3], [5], [4], [11], [10], [9], [6], [7], [2], and [1] provide the terminology and notation for this paper.

1. Parallelism of Subspaces

Let V be a non empty RLS structure and let M, N be Affine subsets of V. We say that M is parallel to N if and only if:

(Def. 1) There exists a vector v of V such that M = N + {v}. One can prove the following propositions:

(1) For every right zeroed non empty RLS structure V holds every Affine subset M of V is parallel to M.

(2) Let V be an add-associative right zeroed right complementable non empty RLS structure and M, N be Affine subsets of V. If M is parallel to N, then N is parallel to M.

(3) Let V be an Abelian add-associative right zeroed right complementable non empty RLS structure and M, L, N be Affine subsets of V. If M is parallel to L and L is parallel to N, then M is parallel to N.

Let V be a non empty loop structure and let M, N be subsets of the carrier of V. The functor M − N yields a subset of V and is defined as follows:

(Def. 2) M − N = {u − v; u ranges over elements of the carrier of V, v ranges

over elements of the carrier of V: u ∈ M ∧ v ∈ N}. Next we state a number of propositions:

(4) For every real linear space V and for all Affine subsets M, N of V holds M − N is Affine.

(5) For every non empty loop structure V and for all subsets M, N of V such that M is empty or N is empty holds M + N is empty.

(6) For every non empty loop structure V and for all non empty subsets M, N of V holds M + N is non empty.

(7) For every non empty loop structure V and for all subsets M, N of V such that M is empty or N is empty holds M − N is empty.

(8) For every non empty loop structure V and for all non empty subsets M, N of V holds M − N is non empty.

(9) Let V be an Abelian add-associative right zeroed right complementable non empty loop structure, M, N be subsets of V, and v be an element of the carrier of V. Then M = N + {v} if and only if M − {v} = N.

(10) Let V be an Abelian add-associative right zeroed right complementable non empty loop structure, M, N be subsets of V, and v be an element of the carrier of V. If v ∈ N, then M + {v} ⊆ M + N.

(11) Let V be an Abelian add-associative right zeroed right complementable non empty loop structure, M, N be subsets of V, and v be an element of the carrier of V. If v ∈ N, then M − {v} ⊆ M − N.

(12) For every real linear space V and for every non empty subset M of V holds 0_{V} ∈ M − M.

(13) Let V be a real linear space, M be a non empty Affine subset of V, and v be a vector of V. If M is Subspace-like and v ∈ M, then M + {v} ⊆ M.

(14) Let V be a real linear space, M be a non empty Affine subset of V, and N_{1}, N_{2} be non empty Affine subsets of V. Suppose N_{1} is Subspace-like and N_{2} is Subspace-like and M is parallel to N_{1} and parallel to N_{2}. Then N_{1} = N_{2}.

(15) Let V be a real linear space, M be a non empty Affine subset of V, and v be a vector of V. If v ∈ M, then 0_{V} ∈ M − {v}.

(16) Let V be a real linear space, M be a non empty Affine subset of V, and v be a vector of V. Suppose v ∈ M. Then there exists a non empty Affine subset N of V such that N = M − {v} and M is parallel to N and N is Subspace-like.

(17) Let V be a real linear space, M be a non empty Affine subset of V, and

u, v be vectors of V. If u ∈ M and v ∈ M, then M − {v} = M − {u}.

(18) For every real linear space V and for every non empty Affine subset M of V holds M − M = ⋃{M − {v}; v ranges over vectors of V: v ∈ M}.

(19) Let V be a real linear space, M be a non empty Affine subset of V, and v be a vector of V. If v ∈ M, then M − {v} = ⋃{M − {u}; u ranges over vectors of V: u ∈ M}.

(20) Let V be a real linear space and M be a non empty Affine subset of V. Then there exists a non empty Affine subset L of V such that L = M −M and L is Subspace-like and M is parallel to L.

2. Orthogonality

Let V be a real unitary space and let W be a subspace of V. The functor Ort CompW yielding a strict subspace of V is defined by:

(Def. 3) The carrier of Ort CompW = {v; v ranges over vectors of V:

⋀_{w: vector of V} (w ∈ W ⇒ w, v are orthogonal)}. Let V be a real unitary space and let M be a non empty subset of V. The

functor Ort CompM yielding a strict subspace of V is defined by: (Def. 4) The carrier of Ort CompM = {v; v ranges over vectors of V:

⋀_{w: vector of V} (w ∈ M ⇒ w, v are orthogonal)}. One can prove the following propositions:

(21) For every real unitary space V and for every subspace W of V holds 0_{V} ∈ Ort CompW.

(22) For every real unitary space V holds Ort Comp0_{V} = Ω_{V} .

(23) For every real unitary space V holds Ort CompΩ_{V} = 0_{V} .

(24) Let V be a real unitary space, W be a subspace of V, and v be a vector of V. If v ≠ 0_{V} , then if v ∈ W, then v ∉ Ort CompW.

(25) For every real unitary space V and for every non empty subset M of V holds M ⊆ the carrier of Ort CompOrt CompM.

(26) Let V be a real unitary space and M, N be non empty subsets of V. If M ⊆ N, then the carrier of Ort CompN ⊆ the carrier of Ort CompM.

(27) Let V be a real unitary space, W be a subspace of V, and M be a non empty subset of V. If M = the carrier of W, then Ort CompM = Ort CompW.

(28) For every real unitary space V and for every non empty subset M of V holds Ort CompM = Ort CompOrt CompOrt CompM.

(29) Let V be a real unitary space and x, y be vectors of V. Then ∥x + y∥^{2} = ∥x∥^{2} + 2 · (x|y) + ∥y∥^{2} and ∥x − y∥^{2} = (∥x∥^{2} − 2 · (x|y)) + ∥y∥^{2}.

(30) Let V be a real unitary space and x, y be vectors of V. If x, y are orthogonal, then ∥x + y∥^{2} = ∥x∥^{2} + ∥y∥^{2}.

(31) For every real unitary space V and for all vectors x, y of V holds ∥x + y∥^{2} + ∥x − y∥^{2} = 2 · ∥x∥^{2} + 2 · ∥y∥^{2}.

(32) Let V be a real unitary space and v be a vector of V. Then there exists a subspace W of V such that the carrier of W = {u; u ranges over vectors of V: (u|v) = 0}.

3. Topology of Real Unitary Space

The scheme SubFamExU deals with a unitary space structure A and a unary predicate P, and states that:

There exists a family F of subsets of A such that for every subset B of the carrier of A holds B ∈ F iff P[B]

for all values of the parameters.

Let V be a real unitary space. The open set family of V yields a family of subsets of V and is defined by the condition (Def. 5).

(Def. 5) Let M be a subset of the carrier of V. Then M ∈ the open set family of V if and only if for every point x of V such that x ∈ M there exists a real number r such that r > 0 and Ball(x, r) ⊆ M.

Next we state several propositions:

(33) Let V be a real unitary space, v be a point of V, and r, p be real numbers. If r p, then Ball(v, r) ⊆ Ball(v, p).

(34) Let V be a real unitary space and v be a point of V. Then there exists a real number r such that r > 0 and Ball(v, r) ⊆ the carrier of V.

(35) Let V be a real unitary space, v, u be points of V, and r be a real number. If u ∈ Ball(v, r), then there exists a real number p such that p > 0 and Ball(u, p) ⊆ Ball(v, r).

(36) Let V be a real unitary space, u, v, w be points of V, and r, p be real numbers. If v ∈ Ball(u, r) ∩ Ball(w, p), then there exists a real number q such that Ball(v, q) ⊆ Ball(u, r) and Ball(v, q) ⊆ Ball(w, p).

(37) Let V be a real unitary space, v be a point of V, and r be a real number. Then Ball(v, r) ∈ the open set family of V.

(38) For every real unitary space V holds the carrier of V ∈ the open set family of V.

(39) Let V be a real unitary space and M, N be subsets of the carrier of V. Suppose M ∈ the open set family of V and N ∈ the open set family of V. Then M ∩ N ∈ the open set family of V.

(40) Let V be a real unitary space and A be a family of subsets of the carrier of V. Suppose A ⊆ the open set family of V. Then ⋃ A ∈ the open set family of V.

(41) For every real unitary space V holds ⟨the carrier of V, the open set

family of V ⟩ is a topological space. Let V be a real unitary space. The functor TopUnitSpaceV yields a topolo-

gical structure and is defined by: (Def. 6) TopUnitSpaceV = ⟨the carrier of V, the open set family of V ⟩.

Let V be a real unitary space. Note that TopUnitSpaceV is topological space-like.

Let V be a real unitary space. One can verify that TopUnitSpaceV is non empty.

We now state a number of propositions:

(42) For every real unitary space V and for every subset M of TopUnitSpaceV such that M = Ω_{V} holds M is open and closed.

(43) For every real unitary space V and for every subset M of TopUnitSpaceV such that M = ∅_{V} holds M is open and closed.

(44) Let V be a real unitary space, v be a vector of V, and r be a real number. If the carrier of V = {0_{V} } and r ≠ 0, then Sphere(v, r) is empty.

(45) Let V be a real unitary space, v be a vector of V, and r be a real number. If the carrier of V ≠ {0_{V} } and r > 0, then Sphere(v, r) is non empty.

(46) Let V be a real unitary space, v be a vector of V, and r be a real number. If r = 0, then Ball(v, r) is empty.

(47) Let V be a real unitary space, v be a vector of V, and r be a real number. If the carrier of V = {0_{V} } and r > 0, then Ball(v, r) = {0_{V} }.

(48) Let V be a real unitary space, v be a vector of V, and r be a real number. Suppose the carrier of V ≠ {0_{V} } and r > 0. Then there exists a vector w of V such that w ≠ v and w ∈ Ball(v, r).

(49) Let V be a real unitary space. Then the carrier of V = the carrier of TopUnitSpaceV and the topology of TopUnitSpaceV = the open set family of V.

(50) Let V be a real unitary space, M be a subset of TopUnitSpaceV, r be a real number, and v be a point of V. If M = Ball(v, r), then M is open.

(51) Let V be a real unitary space and M be a subset of TopUnitSpaceV. Then M is open if and only if for every point v of V such that v ∈ M there exists a real number r such that r > 0 and Ball(v, r) ⊆ M.

(52) Let V be a real unitary space, v_{1}, v_{2} be points of V, and r_{1}, r_{2} be real numbers. Then there exists a point u of V and there exists a real number r such that Ball(v_{1}, r_{1}) ∪ Ball(v_{2}, r_{2}) ⊆ Ball(u, r).

(53) Let V be a real unitary space, M be a subset of TopUnitSpaceV, v be a vector of V, and r be a real number. If M = \overline{Ball}(v, r), then M is closed.

(54) Let V be a real unitary space, M be a subset of TopUnitSpaceV, v be a vector of V, and r be a real number. If M = Sphere(v, r), then M is closed.

References

[1] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23–28, 2003.

[2] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Subspaces and cosets of subspace of real unitary space. Formalized Mathematics, 11(1):1–7, 2003.

[3] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[4] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[5] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[6] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[7] Jan Popiołek. Introduction to Banach and Hilbert spaces - part II. Formalized Mathematics, 2(4):517–521, 1991.

[8] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[9] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[10] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

[11] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[12] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

Armstrong’s Axioms^{1}

William W. Armstrong Dendronic Decisions Ltd Edmonton Yatsuka Nakamura Shinshu University Nagano

Piotr Rudnicki University of Alberta Edmonton

Summary. We present a formalization of the seminal paper by W. W. Armstrong [1] on functional dependencies in relational data bases. The paper is formalized in its entirety including examples and applications. The formalization was done with a routine effort albeit some new notions were defined which simplified formulation of some theorems and proofs.

The definitive reference to the theory of relational databases is [15], where saturated sets are called closed sets. Armstrong’s “axioms” for functional dependencies are still widely taught at all levels of database design, see for instance [13].

MML Identifier: ARMSTRNG.

The articles [21], [10], [28], [11], [24], [30], [32], [31], [18], [3], [9], [7], [26], [22], [4], [23], [25], [14], [20], [2], [5], [29], [8], [6], [17], [16], [27], [19], and [12] provide the notation and terminology for this paper.

1. Preliminaries

The following proposition is true

(1) Let B be a set. Suppose B is ∩-closed. Let X be a set and S be a finite family of subsets of X. If X ∈ B and S ⊆ B, then Intersect(S) ∈ B.

Let us observe that there exists a binary relation which is reflexive, antisymmetric, transitive, and non empty.

One can prove the following proposition

1This work has been supported by NSERC Grant OGP9207 and Shinshu Endowment Fund.

(2) Let R be an antisymmetric transitive non empty binary relation and X be a finite subset of fieldR. If X ≠ ∅, then there exists an element of X which is maximal w.r.t. X, R.

Let X be a set and let R be a binary relation. The functor Maximals_{R}(X) yields a subset of X and is defined by:

(Def. 1) For every set x holds x ∈ Maximals_{R}(X) iff x is maximal w.r.t. X, R.

Let x, X be sets. We say that x is ∩-irreducible in X if and only if: (Def. 2) x ∈ X and for all sets z, y such that z ∈ X and y ∈ X and x = z ∩ y

holds x = z or x = y. We introduce x is ∩-reducible in X as an antonym of x is ∩-irreducible in X.

Let X be a non empty set. The functor ∩-Irreducibles(X) yields a subset of X and is defined by:

(Def. 3) For every set x holds x ∈ ∩-Irreducibles(X) iff x is ∩-irreducible in X.

The scheme FinIntersect deals with a non empty finite set A and a unary predicate P, and states that:

For every set x such that x ∈ A holds P[x] provided the parameters meet the following requirements:

• For every set x such that x is ∩-irreducible in A holds P[x], and

• For all sets x, y such that x ∈ A and y ∈ A and P[x] and P[y] holds P[x ∩ y].

Next we state the proposition

(3) Let X be a non empty finite set and x be an element of X. Then there exists a non empty subset A of X such that x = ⋂ A and for every set s such that s ∈ A holds s is ∩-irreducible in X.

Let X be a set and let B be a family of subsets of X. We say that B is (B1) if and only if:

(Def. 4) X ∈ B.

Let B be a set. We introduce B is (B2) as a synonym of B is ∩-closed. Let X be a set. Observe that there exists a family of subsets of X which is

(B1) and (B2).

The following proposition is true

(4) Let X be a set and B be a non empty family of subsets of X. Suppose B is ∩-closed. Let x be an element of B. Suppose x is ∩-irreducible in B and x ≠ X. Let S be a finite family of subsets of X. If S ⊆ B and x = Intersect(S), then x ∈ S.

Let X, D be non empty sets and let n be a natural number. Observe that every function from X into D^{n} is finite sequence yielding.

Let f be a finite sequence yielding function and let x be a set. Note that f(x) is finite sequence-like.

Let n be a natural number and let p, q be n-tuples of Boolean. The functor p ∧ q yielding a n-tuple of Boolean is defined as follows:

(Def. 5) For every set i such that i ∈ Segn holds (p ∧ q)(i) = p_{i} ∧ q_{i}.

Let us notice that the functor p ∧ q is commutative.

One can prove the following propositions:

(5) For every natural number n and for every n-tuple p of Boolean holds (n -BinarySequence(0)) ∧ p = n -BinarySequence(0).

(6) For every natural number n and for every n-tuple p of Boolean holds ¬(n -BinarySequence(0)) ∧ p = p.

(7) For every natural number i holds (i + 1) -BinarySequence(2^{i}) =

⟨\underbrace{0, …, 0}_{i}⟩^{⌢} ⟨1⟩.

(8) Let n, i be natural numbers. Suppose i < n. Then (n -BinarySequence(2^{i})) (i+1) = 1 and for every natural number j such that j ∈ Segn and j ≠ i+1 holds (n -BinarySequence(2^{i}))(j) = 0.

2. The Relational Model of Data

We consider DB-relationships as systems ⟨ attributes, domains, a relationship ⟩,

where the attributes constitute a finite non empty set, the domains constitute a non-empty many sorted set indexed by the attributes, and the relationship is a subset of ∏ the domains.

3. Dependency Structures

Let X be a set. (Def. 6) A binary relation on 2^{X} is said to be a relation on subsets of X.

We introduce dependency set of X as a synonym of a relation on subsets of X.

Let X be a set. Observe that there exists a dependency set of X which is non empty and finite.

Let X be a set. The functor dependencies(X) yields a dependency set of X and is defined by:

(Def. 7) dependencies(X) = [: 2^{X}, 2^{X} :].

Let X be a set. Observe that dependencies(X) is non empty. A dependency of X is an element of dependencies(X).

Let X be a set and let F be a non empty dependency set of X. We see that the element of F is a dependency of X.

The following three propositions are true:

(9) For all sets X, x holds x ∈ dependencies(X) iff there exist subsets a, b of X such that x = ⟨⟨a, b⟩⟩.

(10) For all sets X, x and for every dependency set F of X such that x ∈ F there exist subsets a, b of X such that x = ⟨⟨a, b⟩⟩.

(11) For every set X and for every dependency set F of X holds every subset

of F is a dependency set of X. Let R be a DB-relationship and let A, B be subsets of the attributes of R.

The predicate A →_{R} B is defined by: (Def. 8) For all elements f, g of the relationship of R such that f↾A = g↾A holds

f↾B = g↾B. We introduce (A, B) holds in R as a synonym of A →_{R} B.

In the sequel R denotes a DB-relationship and A, B denote subsets of the attributes of R.

Let us consider R. The functor dependency-structure(R) yields a dependency set of the attributes of R and is defined as follows:

(Def. 9) dependency-structure(R) = {⟨⟨A, B⟩⟩: A →_{R} B}.

One can prove the following proposition

(12) For every DB-relationship R and for all subsets A, B of the attributes of R holds ⟨⟨A, B⟩⟩∈ dependency-structure(R) iff A →_{R} B.

4. Full Families of Dependencies

Let X be a set and let P, Q be dependencies of X. The predicate P Q is defined by:

(Def. 10) P_{1} ⊆ Q_{1} and Q_{2} ⊆ P_{2}. Let us note that the predicate P Q is reflexive. We introduce Q P and also P is at least as informative as Q, as synonyms of P Q.

The following propositions are true:

(13) For every set X and for all dependencies P, Q of X such that P Q and Q P holds P = Q.

(14) For every set X and for all dependencies P, Q, S of X such that P Q

and Q S holds P S. Let X be a set and let A, B be subsets of X. Then ⟨⟨A, B⟩⟩ is a dependency

of X.

We now state the proposition

(15) For every set X and for all subsets A, B, A′, B′of X holds ⟨⟨A, B⟩⟩ ⟨⟨A′,

B′⟩⟩ iff A ⊆ A′ and B′⊆ B. Let X be a set. The functor Dependencies-OrderX yielding a binary relation

on dependencies(X) is defined as follows:

(Def. 11) Dependencies-OrderX = {⟨⟨P, Q⟩⟩; P ranges over dependencies of X, Q

ranges over dependencies of X: P Q}. We now state four propositions:

(16) For all sets X, x holds x ∈ Dependencies-OrderX iff there exist dependencies P, Q of X such that x = ⟨⟨P, Q⟩⟩ and P Q.

(17) For every set X holds domDependencies-OrderX = [: 2^{X}, 2^{X} :].

(18) For every set X holds rngDependencies-OrderX = [: 2^{X}, 2^{X} :].

(19) For every set X holds fieldDependencies-OrderX = [: 2^{X}, 2^{X} :]. Let X be a set. Note that Dependencies-OrderX is non empty and Dependencies-OrderX is ordering. Let X be a set and let F be a dependency set of X. We say that F is (F1)

if and only if: (Def. 12) For every subset A of X holds ⟨⟨A, A⟩⟩∈ F.

We introduce F is (DC2) as a synonym of F is (F1). We introduce F is (F2) and F is (DC1) as synonyms of F is transitive.

The following proposition is true

(20) Let X be a set and F be a dependency set of X. Then F is (F2) if and only if for all subsets A, B, C of X such that ⟨⟨A, B⟩⟩∈ F and ⟨⟨B, C⟩⟩∈ F holds ⟨⟨A, C⟩⟩∈ F.

Let X be a set and let F be a dependency set of X. We say that F is (F3) if and only if:

(Def. 13) For all subsets A, B, A′, B′of X such that ⟨⟨A, B⟩⟩∈ F and ⟨⟨A, B⟩⟩ ⟨⟨A′,

B′⟩⟩ holds ⟨⟨A′, B′⟩⟩∈ F. We say that F is (F4) if and only if:

(Def. 14) For all subsets A, B, A′, B′of X such that ⟨⟨A, B⟩⟩∈ F and ⟨⟨A′, B′⟩⟩∈ F

holds ⟨⟨A ∪ A′, B ∪ B′⟩⟩∈ F. The following proposition is true

(21) For every set X holds dependencies(X) is (F1), (F2), (F3), and (F4). Let X be a set. Observe that there exists a dependency set of X which is

(F1), (F2), (F3), (F4), and non empty.

Let X be a set and let F be a dependency set of X. We say that F is full family if and only if:

(Def. 15) F is (F1), (F2), (F3), and (F4).

Let X be a set. One can verify that there exists a dependency set of X which is full family.

Let X be a set. A Full family of X is a full family dependency set of X. We now state the proposition

(22) For every finite set X holds every dependency set of X is finite.

Let X be a finite set. Observe that there exists a Full family of X which is finite and every dependency set of X is finite.

Let X be a set. Note that every dependency set of X which is full family is also (F1), (F2), (F3), and (F4) and every dependency set of X which is (F1), (F2), (F3), and (F4) is also full family.

Let X be a set and let F be a dependency set of X. We say that F is (DC3) if and only if:

(Def. 16) For all subsets A, B of X such that B ⊆ A holds ⟨⟨A, B⟩⟩∈ F.

Let X be a set. Observe that every dependency set of X which is (F1) and (F3) is also (DC3) and every dependency set of X which is (DC3) and (F2) is also (F1) and (F3).

Let X be a set. Observe that there exists a dependency set of X which is (DC3), (F2), (F4), and non empty.

We now state two propositions:

(23) For every set X and for every dependency set F of X such that F is (DC3) and (F2) holds F is (F1) and (F3).

(24) For every set X and for every dependency set F of X such that F is

(F1) and (F3) holds F is (DC3). Let X be a set. Observe that every dependency set of X which is (F1) is

also non empty.

The following propositions are true:

(25) For every DB-relationship R holds dependency-structure(R) is full family.

(26) Let X be a set and K be a subset of X. Then {⟨⟨A, B⟩⟩; A ranges over subsets of X, B ranges over subsets of X: K ⊆ A ∨ B ⊆ A} is a Full family of X.

5. Maximal Elements of Full Families

Let X be a set and let F be a dependency set of X. The functor Maximals(F) yielding a dependency set of X is defined as follows:

(Def. 17) Maximals(F) = Maximals_{Dependencies-Order X}(F).

We now state the proposition

(27) For every set X and for every dependency set F of X holds

Maximals(F) ⊆ F. Let X be a set, let F be a dependency set of X, and let x, y be sets. The

predicate x ↗_{F} y is defined as follows: (Def. 18) ⟨⟨x, y⟩⟩∈ Maximals(F).

One can prove the following two propositions:

(28) Let X be a finite set, P be a dependency of X, and F be a dependency set of X. If P ∈ F, then there exist subsets A, B of X such that ⟨⟨A, B⟩⟩∈ Maximals(F) and ⟨⟨A, B⟩⟩ P.

(29) Let X be a set, F be a dependency set of X, and A, B be subsets of X.

Then A ↗_{F} B if and only if the following conditions are satisfied:

(i) ⟨⟨A, B⟩⟩∈ F, and (ii) it is not true that there exist subsets A′, B′of X such that ⟨⟨A′, B′⟩⟩∈ F

and ⟨⟨A, B⟩⟩ ⟨⟨A′, B′⟩⟩ with A ≠ A′ or B ≠ B′. Let X be a set and let M be a dependency set of X. We say that M is (M1)

if and only if: (Def. 19) For every subset A of X there exist subsets A′, B′ of X such that ⟨⟨A′,

B′⟩⟩ ⟨⟨A, A⟩⟩ and ⟨⟨A′, B′⟩⟩∈ M. We say that M is (M2) if and only if:

(Def. 20) For all subsets A, B, A′, B′of X such that ⟨⟨A, B⟩⟩∈ M and ⟨⟨A′, B′⟩⟩∈ M

and ⟨⟨A, B⟩⟩ ⟨⟨A′, B′⟩⟩ holds A = A′ and B = B′. We say that M is (M3) if and only if:

(Def. 21) For all subsets A, B, A′, B′of X such that ⟨⟨A, B⟩⟩∈ M and ⟨⟨A′, B′⟩⟩∈ M

and A′⊆ B holds B′⊆ B. We now state two propositions:

(30) For every finite non empty set X and for every Full family F of X holds Maximals(F) is (M1), (M2), and (M3).

(31) Let X be a finite set and M, F be dependency sets of X. Suppose that (i) M is (M1), (M2), and (M3), and (ii) F = {⟨⟨A, B⟩⟩; A ranges over subsets of X, B ranges over subsets of X:

⋁_{A′,B′: subset of X} (⟨⟨A′, B′⟩⟩ ⟨⟨A, B⟩⟩ ∧ ⟨⟨A′, B′⟩⟩∈ M)}. Then M = Maximals(F) and F is full family and for every Full family G of X such that M = Maximals(G) holds G = F.

Let X be a non empty finite set and let F be a Full family of X. Note that Maximals(F) is non empty.

Next we state the proposition

(32) Let X be a finite set, F be a dependency set of X, and K be a subset of X. Suppose F = {⟨⟨A, B⟩⟩; A ranges over subsets of X, B ranges over subsets of X: K ⊆ A ∨ B ⊆ A}. Then {⟨⟨K, X⟩⟩}∪{⟨⟨A, A⟩⟩; A ranges over subsets of X: K ⊈ A} = Maximals(F).

6. Saturated Subsets of Attributes

Let X be a set and let F be a dependency set of X. The functor saturated-subsets(F) yields a family of subsets of X and is

defined as follows:

(Def. 22) saturated-subsets(F) =

{B; B ranges over subsets of X: ⋁_{A: subset of X} A ↗_{F} B}. We introduce closed-attribute-subset(F) as a synonym of saturated-subsets(F).

Let X be a set and let F be a finite dependency set of X. Observe that saturated-subsets(F) is finite.

Next we state two propositions:

(33) Let X, x be sets and F be a dependency set of X. Then x ∈ saturated-subsets(F) if and only if there exist subsets B, A of X such that x = B and A ↗_{F} B.

(34) For every finite non empty set X and for every Full family F of X holds

saturated-subsets(F) is (B1) and (B2). Let X be a set and let B be a set. The functor (B)-enclosed in X yields a

dependency set of X and is defined as follows: (Def. 23) (B)-enclosed in X = {⟨⟨a, b⟩⟩; a ranges over subsets of X, b ranges over

subsets of X: ⋀_{c: set} (c ∈ B ∧ a ⊆ c ⇒ b ⊆ c)}. The following three propositions are true:

(35) For every set X and for every family B of subsets of X and for every dependency set F of X holds (B)-enclosed in X is full family.

(36) For every finite non empty set X and for every family B of subsets of X holds B ⊆ saturated-subsets((B)-enclosed in X).

(37) Let X be a finite non empty set and B be a family of subsets of X. Suppose B is (B1) and (B2). Then B = saturated-subsets((B)-enclosed in X) and for every Full family G of X such that B = saturated-subsets(G) holds G = (B)-enclosed in X.

Let X be a set and let F be a dependency set of X. The functor (F)-enclosure yielding a family of subsets of X is defined as follows:

(Def. 24) (F)-enclosure = {b; b ranges over subsets of X: ⋀_{A,B: subset of X} (⟨⟨A,

B⟩⟩∈ F ∧ A ⊆ b ⇒ B ⊆ b)}. We now state two propositions:

(38) For every finite non empty set X and for every dependency set F of X holds (F)-enclosure is (B1) and (B2).

(39) Let X be a finite non empty set and F be a dependency set of X. Then F ⊆ ((F)-enclosure)-enclosed in X and for every dependency set G of X such that F ⊆ G and G is full family holds ((F)-enclosure)-enclosed in X ⊆ G.

Let X be a finite non empty set and let F be a dependency set of X. The functor dependency-closure(F) yields a Full family of X and is defined by:

(Def. 25) F ⊆ dependency-closure(F) and for every dependency set G of X such that F ⊆ G and G is full family holds dependency-closure(F) ⊆ G.

Next we state four propositions:

(40) For every finite non empty set X and for every dependency set F of X holds dependency-closure(F) = ((F)-enclosure)-enclosed in X.

(41) Let X be a set, K be a subset of X, and B be a family of subsets of X. If B = {X} ∪ {A; A ranges over subsets of X: K ⊈ A}, then B is (B1) and (B2).

(42) Let X be a finite non empty set, F be a dependency set of X, and K be a subset of X. Suppose F = {⟨⟨A, B⟩⟩; A ranges over subsets of X, B ranges over subsets of X: K ⊆ A ∨ B ⊆ A}. Then {X} ∪ {B; B ranges over subsets of X: K ⊈ B} = saturated-subsets(F).

(43) Let X be a finite set, F be a dependency set of X, and K be a subset of X. Suppose F = {⟨⟨A, B⟩⟩; A ranges over subsets of X, B ranges over subsets of X: K ⊆ A ∨ B ⊆ A}. Then {X} ∪ {B; B ranges over subsets of X: K ⊈ B} = saturated-subsets(F).

Let X, G be sets and let B be a family of subsets of X. We say that G is generator set of B if and only if:

(Def. 26) G ⊆ B and B = {Intersect(S); S ranges over families of subsets of X:

S ⊆ G}. We now state four propositions:

(44) For every finite non empty set X holds every family G of subsets of X is generator set of saturated-subsets((G)-enclosed in X).

(45) Let X be a finite non empty set and F be a Full family of X. Then there exists a family G of subsets of X such that G is generator set of saturated-subsets(F) and F = (G)-enclosed in X.

(46) Let X be a set and B be a non empty finite family of subsets of X. If B is (B1) and (B2), then ∩-Irreducibles(B) is generator set of B.

(47) Let X, G be sets and B be a non empty finite family of subsets of X. If B is (B1) and (B2) and G is generator set of B, then ∩-Irreducibles(B) ⊆ G ∪ {X}.

7. Justification of the Axioms

One can prove the following proposition

(48) Let X be a non empty finite set and F be a Full family of X. Then there exists a DB-relationship R such that the attributes of R = X and for every element a of X holds (the domains of R)(a) = ℤ and F = dependency-structure(R).

8. Structure of the Family of Candidate Keys

Let X be a set and let F be a dependency set of X. The functor candidate-keys(F) yields a family of subsets of X and is defined

by: (Def. 27) candidate-keys(F) = {A; A ranges over subsets of X: ⟨⟨A, X⟩⟩ ∈

Maximals(F)}. One can prove the following proposition

(49) Let X be a finite set, F be a dependency set of X, and K be a subset of X. Suppose F = {⟨⟨A, B⟩⟩; A ranges over subsets of X, B ranges over subsets of X: K ⊆ A ∨ B ⊆ A}. Then candidate-keys(F) = {K}.

Let X be a set. We introduce X is (C1) as an antonym of X is empty. Let X be a set. We say that X is without proper subsets if and only if:

(Def. 28) For all sets x, y such that x ∈ X and y ∈ X and x ⊆ y holds x = y.

We introduce X is (C2) as a synonym of X is without proper subsets.

We now state four propositions:

(50) For every DB-relationship R holds candidate-keys(dependency-structure(R)) is (C1) and (C2).

(51) Let X be a finite set and C be a family of subsets of X. If C is (C1) and (C2), then there exists a Full family F of X such that C = candidate-keys(F).

(52) Let X be a finite set, C be a family of subsets of X, and B be a set. Suppose C is (C1) and (C2) and B = {b; b ranges over subsets of X: ⋀_{K: subset of X} (K ∈ C ⇒ K ⊈ b)}. Then C = candidate-keys((B)-enclosed in X).

(53) Let X be a non empty finite set and C be a family of subsets of X. Suppose C is (C1) and (C2). Then there exists a DBrelationship R such that the attributes of R = X and C = candidate-keys(dependency-structure(R)).

9. Applications

Let X be a set and let F be a dependency set of X. We say that F is (DC4) if and only if:

(Def. 29) For all subsets A, B, C of X such that ⟨⟨A, B⟩⟩ ∈ F and ⟨⟨A, C⟩⟩ ∈ F

holds ⟨⟨A, B ∪ C⟩⟩∈ F. We say that F is (DC5) if and only if:

(Def. 30) For all subsets A, B, C, D of X such that ⟨⟨A, B⟩⟩ ∈ F and ⟨⟨B ∪ C, D⟩⟩∈ F holds ⟨⟨A ∪ C, D⟩⟩∈ F.

We say that F is (DC6) if and only if: (Def. 31) For all subsets A, B, C of X such that ⟨⟨A, B⟩⟩∈ F holds ⟨⟨A∪C, B⟩⟩∈ F.

One can prove the following propositions:

(54) Let X be a set and F be a dependency set of X. Then F is (F1), (F2), (F3), and (F4) if and only if F is (F2), (DC3), and (F4).

(55) Let X be a set and F be a dependency set of X. Then F is (F1), (F2), (F3), and (F4) if and only if F is (DC1), (DC3), and (DC4).

(56) Let X be a set and F be a dependency set of X. Then F is (F1), (F2),

(F3), and (F4) if and only if F is (DC2), (DC5), and (DC6). Let X be a set and let F be a dependency set of X. The functor characteristic(F) is defined as follows:

(Def. 32) characteristic(F) = {A; A ranges over subsets of X: ⋁_{a,b: subset of X} (⟨⟨a,

b⟩⟩∈ F ∧ a ⊆ A ∧ b ⊈ A)}. Next we state several propositions:

(57) Let X, A be sets and F be a dependency set of X. Suppose A ∈ characteristic(F). Then A is a subset of X and there exist subsets a, b of X such that ⟨⟨a, b⟩⟩∈ F and a ⊆ A and b ⊈ A.

(58) Let X be a set, A be a subset of X, and F be a dependency set of X. If there exist subsets a, b of X such that ⟨⟨a, b⟩⟩ ∈ F and a ⊆ A and b ⊈ A, then A ∈ characteristic(F).

(59) Let X be a finite non empty set and F be a dependency set of X. Then

(i) for all subsets A, B of X holds ⟨⟨A, B⟩⟩ ∈ dependency-closure(F) iff for every subset a of X such that A ⊆ a and B ⊈ a holds a ∈ characteristic(F), and

(ii) saturated-subsets(dependency-closure(F)) = 2^{X} \ characteristic(F).

(60) For every finite non empty set X and for all dependency sets F, G of X such that characteristic(F) = characteristic(G) holds dependency-closure(F) = dependency-closure(G).

(61) For every non empty finite set X and for every dependency set F of X

holds characteristic(F) = characteristic(dependency-closure(F)). Let A be a set, let K be a set, and let F be a dependency set of A. We say

that K is prime implicant of F with no complemented variables if and only if the conditions (Def. 33) are satisfied.

(Def. 33)(i) For every subset a of A such that K ⊆ a and a ≠ A holds a ∈

characteristic(F), and (ii) for every set k such that k ⊂ K there exists a subset a of A such that

k ⊆ a and a ≠ A and a ∉ characteristic(F). The following proposition is true 50 william w. armstrong et al.

(62) Let X be a finite non empty set, F be a dependency set of X, and K be a subset of X. Then K ∈ candidate-keys(dependency-closure(F)) if and only if K is prime implicant of F with no complemented variables.

References

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[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

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armstrong’s axioms 51

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Mathematics, 1(1):85–89, 1990.

Received October 25, 2002

Convex Sets and Convex Combinations

Noboru Endou

Gifu National College of Technology Takashi Mitsuishi Miyagi University

Yasunari Shidama Shinshu University Nagano

Summary. Convexity is one of the most important concepts in a study of analysis. Especially, it has been applied around the optimization problem widely. Our purpose is to define the concept of convexity of a set on Mizar, and to develop the generalities of convex analysis. The construction of this article is as follows: Convexity of the set is defined in the section 1. The section 2 gives the definition of convex combination which is a kind of the linear combination and related theorems are proved there. In section 3, we define the convex hull which is an intersection of all convex sets including a given set. The last section is some theorems which are necessary to compose this article.

MML Identifier: CONVEX1.

The notation and terminology used in this paper are introduced in the following articles: [13], [12], [17], [9], [10], [3], [1], [8], [4], [2], [16], [15], [14], [5], [11], [6], and [7].

1. Convex Sets

Let V be a non empty RLS structure, let M be a subset of V, and let r be a real number. The functor r · M yielding a subset of V is defined by:

(Def. 1) r · M = {r · v; v ranges over elements of the carrier of V: v ∈ M}.

Let V be a non empty RLS structure and let M be a subset of V. We say that M is convex if and only if:

(Def. 2) For all vectors u, v of V and for every real number r such that 0 < r and r < 1 and u ∈ M and v ∈ M holds r · u + (1 − r) · v ∈ M.

We now state a number of propositions:

(1) Let V be a real linear space-like non empty RLS structure, M be a subset of V, and r be a real number. If M is convex, then r · M is convex.

(2) Let V be an Abelian add-associative real linear space-like non empty RLS structure and M, N be subsets of V. If M is convex and N is convex, then M + N is convex.

(3) For every real linear space V and for all subsets M, N of V such that M is convex and N is convex holds M − N is convex.

(4) Let V be a non empty RLS structure and M be a subset of V. Then M is convex if and only if for every real number r such that 0 < r and r < 1 holds r · M + (1 − r) · M ⊆ M.

(5) Let V be an Abelian non empty RLS structure and M be a subset of V. Suppose M is convex. Let r be a real number. If 0 < r and r < 1, then (1 − r) · M + r · M ⊆ M.

(6) Let V be an Abelian add-associative real linear space-like non empty RLS structure and M, N be subsets of V. Suppose M is convex and N is convex. Let r be a real number. Then r · M + (1 − r) · N is convex.

(7) Let V be a real linear space, M be a subset of V, and v be a vector of V. Then M is convex if and only if v + M is convex.

(8) For every real linear space V holds Up(0_{V} ) is convex.

(9) For every real linear space V holds Up(Ω_{V}) is convex.

(10) For every non empty RLS structure V and for every subset M of V such that M = ∅ holds M is convex.

(11) Let V be an Abelian add-associative real linear space-like non empty RLS structure, M_{1}, M_{2} be subsets of V, and r_{1}, r_{2} be real numbers. If M_{1} is convex and M_{2} is convex, then r_{1} · M_{1} + r_{2} · M_{2} is convex.

(12) Let V be a real linear space-like non empty RLS structure, M be a subset of V, and r_{1}, r_{2} be real numbers. Then (r_{1} + r_{2}) · M ⊆ r_{1} · M + r_{2} · M.

(13) Let V be a real linear space, M be a subset of V, and r_{1}, r_{2} be real numbers. If r_{1} 0 and r_{2} 0 and M is convex, then r_{1} · M + r_{2} · M ⊆ (r_{1} + r_{2}) · M.

(14) Let V be an Abelian add-associative real linear space-like non empty RLS structure, M_{1}, M_{2}, M_{3} be subsets of V, and r_{1}, r_{2}, r_{3} be real numbers. If M_{1} is convex and M_{2} is convex and M_{3} is convex, then r_{1} · M_{1} + r_{2} · M_{2} + r_{3} · M_{3} is convex.

(15) Let V be a non empty RLS structure and F be a family of subsets of V. Suppose that for every subset M of V such that M ∈ F holds M is convex. Then ⋂ F is convex.

(16) For every non empty RLS structure V and for every subset M of V such

that M is Affine holds M is convex. Let V be a non empty RLS structure. Observe that there exists a subset of

V which is convex.

Let V be a non empty RLS structure. Note that there exists a subset of V which is empty and convex.

Let V be a non empty RLS structure. One can check that there exists a subset of V which is non empty and convex.

The following four propositions are true:

(17) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, v be a vector of V, and r be a real number. If M = {u; u ranges over vectors of V: (u|v) r}, then M is convex.

(18) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, v be a vector of V, and r be a real number. If M = {u; u ranges over vectors of V: (u|v) > r}, then M is convex.

(19) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, v be a vector of V, and r be a real number. If M = {u; u ranges over vectors of V: (u|v) r}, then M is convex.

(20) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, v be a vector of V, and r be a real number. If M = {u; u ranges over vectors of V: (u|v) < r}, then M is convex.

2. Convex Combinations

Let V be a real linear space and let L be a linear combination of V. We say that L is convex if and only if the condition (Def. 3) is satisfied.

(Def. 3) There exists a finite sequence F of elements of the carrier of V such that (i) F is one-to-one, (ii) rngF = the support of L, and (iii) there exists a finite sequence f of elements of ℝ such that lenf = lenF

and ∑ f = 1 and for every natural number n such that n ∈ domf holds f(n) = L(F (n)) and f(n) 0.

One can prove the following propositions:

(21) Let V be a real linear space and L be a linear combination of V. If L is convex, then the support of L ≠ ∅.

(22) Let V be a real linear space, L be a linear combination of V, and v be a vector of V. If L is convex and L(v) 0, then v ∉ the support of L.

(23) For every real linear space V and for every linear combination L of V such that L is convex holds L ≠ 0_{LC_{V}} .

(24) Let V be a real linear space, v be a vector of V, and L be a linear combination of {v}. If L is convex, then L(v) = 1 and ∑ L = L(v) · v.

(25) Let V be a real linear space, v_{1}, v_{2} be vectors of V, and L be a linear combination of {v_{1}, v_{2}}. Suppose v_{1} ≠ v_{2} and L is convex. Then L(v_{1}) + L(v_{2}) = 1 and L(v_{1}) 0 and L(v_{2}) 0 and ∑ L = L(v_{1}) · v_{1} + L(v_{2}) · v_{2}.

(26) Let V be a real linear space, v_{1}, v_{2}, v_{3} be vectors of V, and L be a linear combination of {v_{1}, v_{2}, v_{3}}. Suppose v_{1} ≠ v_{2} and v_{2} ≠ v_{3} and v_{3} ≠ v_{1} and L is convex. Then L(v_{1}) + L(v_{2}) + L(v_{3}) = 1 and L(v_{1}) 0 and L(v_{2}) 0 and L(v_{3}) 0 and ∑ L = L(v_{1}) · v_{1} + L(v_{2}) · v_{2} + L(v_{3}) · v_{3}.

(27) Let V be a real linear space, v be a vector of V, and L be a linear combination of V. If L is convex and the support of L = {v}, then L(v) = 1.

(28) Let V be a real linear space, v_{1}, v_{2} be vectors of V, and L be a linear combination of V. Suppose L is convex and the support of L = {v_{1}, v_{2}} and v_{1} ≠ v_{2}. Then L(v_{1}) + L(v_{2}) = 1 and L(v_{1}) 0 and L(v_{2}) 0.

(29) Let V be a real linear space, v_{1}, v_{2}, v_{3} be vectors of V, and L be a linear combination of V. Suppose L is convex and the support of L = {v_{1}, v_{2}, v_{3}} and v_{1} ≠ v_{2} and v_{2} ≠ v_{3} and v_{3} ≠ v_{1}. Then L(v_{1}) + L(v_{2}) + L(v_{3}) = 1 and L(v_{1}) 0 and L(v_{2}) 0 and L(v_{3}) 0 and ∑ L = L(v_{1}) · v_{1} + L(v_{2}) · v_{2} + L(v_{3}) · v_{3}.

3. Convex Hull

In this article we present several logical schemes. The scheme SubFamExRLS deals with an RLS structure A and a unary predicate P, and states that:

There exists a family F of subsets of A such that for every subset B of the carrier of A holds B ∈ F iff P[B]

for all values of the parameters.

The scheme SubFamExRLS2 deals with an RLS structure A and a unary predicate P, and states that:

There exists a family F of subsets of A such that for every subset B of A holds B ∈ F iff P[B]

for all values of the parameters.

Let V be a non empty RLS structure and let M be a subset of V. The functor Convex-FamilyM yields a family of subsets of V and is defined as follows:

(Def. 4) For every subset N of V holds N ∈ Convex-FamilyM iff N is convex

and M ⊆ N. Let V be a non empty RLS structure and let M be a subset of V. The

functor convM yields a convex subset of V and is defined by: (Def. 5) convM = ⋂ Convex-FamilyM.

The following proposition is true (30) Let V be a non empty RLS structure, M be a subset of V, and N be a

convex subset of V. If M ⊆ N, then convM ⊆ N.

4. Miscellaneous

One can prove the following propositions:

(31) Let p be a finite sequence and x, y, z be sets. Suppose p is one-to-one and rngp = {x, y, z} and x ≠ y and y ≠ z and z ≠ x. Then p = ⟨x, y, z⟩ or p = ⟨x, z, y⟩ or p = ⟨y, x, z⟩ or p = ⟨y, z, x⟩ or p = ⟨z, x, y⟩ or p = ⟨z, y, x⟩.

(32) For every real linear space-like non empty RLS structure V and for every subset M of V holds 1 · M = M.

(33) For every non empty RLS structure V and for every empty subset M of V and for every real number r holds r · M = ∅.

(34) For every real linear space V and for every non empty subset M of V holds 0 · M = {0_{V} }.

(35) For every right zeroed non empty loop structure V and for every subset M of V holds M + {0_{V} } = M.

(36) For every add-associative non empty loop structure V and for all subsets M_{1}, M_{2}, M_{3} of V holds (M_{1} + M_{2}) + M_{3} = M_{1} + (M_{2} + M_{3}).

(37) Let V be a real linear space-like non empty RLS structure, M be a subset of V, and r_{1}, r_{2} be real numbers. Then r_{1} · (r_{2} · M) = (r_{1} · r_{2}) · M.

(38) Let V be a real linear space-like non empty RLS structure, M_{1}, M_{2} be subsets of V, and r be a real number. Then r ·(M_{1} +M_{2}) = r ·M_{1} +r ·M_{2}.

(39) Let V be a non empty RLS structure, M, N be subsets of V, and r be a real number. If M ⊆ N, then r · M ⊆ r · N.

(40) For every non empty loop structure V and for every empty subset M of V and for every subset N of V holds M + N = ∅.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[6] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23–28, 2003.

[7] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Topology of real unitary space. Formalized Mathematics, 11(1):33–38, 2003.

[8] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

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Quotient Vector Spaces and Functionals^{1}

Jarosław Kotowicz University of Białystok

Summary. The article presents well known facts about quotient vector spaces and functionals (see [8]). There are repeated theorems and constructions with either weaker assumptions or in more general situations (see [11], [7], [10]). The construction of coefficient functionals and non-degenerate functional in quotient vector space generated by functional in the given vector space are the only new things which are done.

MML Identifier: VECTSP10.

The articles [15], [5], [21], [13], [3], [1], [20], [2], [17], [7], [22], [4], [6], [14], [19], [12], [18], [16], and [9] provide the notation and terminology for this paper.

1. Auxiliary Facts about Double Loops and Vector Spaces

The following proposition is true

(1) Let K be an add-associative right zeroed right complementable left distributive left unital non empty double loop structure and a be an element of the carrier of K. Then (− 1_{K}) · a = − a.

Let K be a double loop structure. The functor StructVectSp(K) yields a strict vector space structure over K and is defined as follows:

(Def. 1) StructVectSp(K) = ⟨the carrier of K, the addition of K, the zero of K,

the multiplication of K⟩. Let K be a non empty double loop structure. Note that StructVectSp(K) is

non empty.

Let K be an Abelian non empty double loop structure. One can verify that StructVectSp(K) is Abelian.

1This work has been partially supported by TRIAL-SOLUTION grant IST-2001-35447 and SPUB-M grant of KBN.

Let K be an add-associative non empty double loop structure. Note that StructVectSp(K) is add-associative.

Let K be a right zeroed non empty double loop structure.

Note that StructVectSp(K) is right zeroed.

Let K be a right complementable non empty double loop structure. Observe that StructVectSp(K) is right complementable.

Let K be an associative left unital distributive non empty double loop structure. One can check that StructVectSp(K) is vector space-like.

Let K be a non degenerated non empty double loop structure. Note that StructVectSp(K) is non trivial.

Let K be a non degenerated non empty double loop structure. Note that there exists a non empty vector space structure over K which is non trivial.

Let K be an add-associative right zeroed right complementable non empty double loop structure. Observe that there exists a non empty vector space structure over K which is add-associative, right zeroed, right complementable, and strict.

Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure. One can check that there exists a non empty vector space structure over K which is add-associative, right zeroed, right complementable, vector space-like, and strict.

Let K be an Abelian add-associative right zeroed right complementable associative left unital distributive non degenerated non empty double loop structure. One can verify that there exists a non empty vector space structure over K which is Abelian, add-associative, right zeroed, right complementable, vector space-like, strict, and non trivial.

Next we state a number of propositions:

(2) Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, a be an element of the carrier of K, V be an add-associative right zeroed right complementable vector space-like non empty vector space structure over K, and v be a vector of V. Then 0_{K} · v = 0_{V} and a · 0_{V} = 0_{V} .

(3) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, S, T be subspaces of V, and v be a vector of V. If S ∩ T = 0_{V} and v ∈ S and v ∈ T, then v = 0_{V} .

(4) Let K be a field, V be a vector space over K, x be a set, and v be a vector of V. Then x ∈ Lin({v}) if and only if there exists an element a of the carrier of K such that x = a · v.

(5) Let K be a field, V be a vector space over K, v be a vector of V, and a, b be scalars of V. If v ≠ 0_{V} and a · v = b · v, then a = b.

(6) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, and W_{1}, W_{2} be subspaces of V. Suppose V is the direct sum of W_{1} and W_{2}. Let v, v_{1}, v_{2} be vectors of V. If v_{1} ∈ W_{1} and v_{2} ∈ W_{2} and v = v_{1} + v_{2}, then v_{⟨⟨W_{1},W_{2}⟩⟩} = ⟨⟨v_{1}, v_{2}⟩⟩.

(7) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, and W_{1}, W_{2} be subspaces of V. Suppose V is the direct sum of W_{1} and W_{2}. Let v, v_{1}, v_{2} be vectors of V. If v_{⟨⟨W_{1},W_{2}⟩⟩} = ⟨⟨v_{1}, v_{2}⟩⟩, then v = v_{1} + v_{2}.

(8) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, and W_{1}, W_{2} be subspaces of V. Suppose V is the direct sum of W_{1} and W_{2}. Let v, v_{1}, v_{2} be vectors of V. If v_{⟨⟨W_{1},W_{2}⟩⟩} = ⟨⟨v_{1}, v_{2}⟩⟩, then v_{1} ∈ W_{1} and v_{2} ∈ W_{2}.

(9) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, and W_{1}, W_{2} be subspaces of V. Suppose V is the direct sum of W_{1} and W_{2}. Let v, v_{1}, v_{2} be vectors of V. If v_{⟨⟨W_{1},W_{2}⟩⟩} = ⟨⟨v_{1}, v_{2}⟩⟩, then v_{⟨⟨W_{2},W_{1}⟩⟩} = ⟨⟨v_{2}, v_{1}⟩⟩.

(10) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, and W_{1}, W_{2} be subspaces of V. Suppose V is the direct sum of W_{1} and W_{2}. Let v be a vector of V. If v ∈ W_{1}, then v_{⟨⟨W_{1},W_{2}⟩⟩} = ⟨⟨v, 0_{V} ⟩⟩.

(11) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, and W_{1}, W_{2} be subspaces of V. Suppose V is the direct sum of W_{1} and W_{2}. Let v be a vector of V. If v ∈ W_{2}, then v_{⟨⟨W_{1},W_{2}⟩⟩} = ⟨⟨0_{V} , v⟩⟩.

(12) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, V_{1} be a subspace of V, W_{1} be a subspace of V_{1}, and v be a vector of V. If v ∈ W_{1}, then v is a vector of V_{1}.

(13) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, V_{1}, V_{2}, W be subspaces of V, and W_{1}, W_{2} be subspaces of W. If W_{1} = V_{1} and W_{2} = V_{2}, then W_{1} + W_{2} = V_{1} + V_{2}.

(14) Let K be a field, V be a vector space over K, W be a subspace of V, v be a vector of V, and w be a vector of W. If v = w, then Lin({w}) = Lin({v}).

(15) Let K be a field, V be a vector space over K, v be a vector of V, and X be a subspace of V. Suppose v ∉ X. Let y be a vector of X +Lin({v}) and W be a subspace of X +Lin({v}). If v = y and W = X, then X +Lin({v}) is the direct sum of W and Lin({y}).

(16) Let K be a field, V be a vector space over K, v be a vector of V, X be a subspace of V, y be a vector of X + Lin({v}), and W be a subspace of X + Lin({v}). If v = y and X = W and v ∉ X, then y_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨0_{W} , y⟩⟩.

(17) Let K be a field, V be a vector space over K, v be a vector of V, X be a subspace of V, y be a vector of X + Lin({v}), and W be a subspace of X + Lin({v}). Suppose v = y and X = W and v ∉ X. Let w be a vector of X + Lin({v}). If w ∈ X, then w_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨w, 0_{V} ⟩⟩.

(18) Let K be an add-associative right zeroed right complementable associative Abelian left unital distributive non empty double loop structure, V be a vector space over K, v be a vector of V, and W_{1}, W_{2} be subspaces of V. Then there exist vectors v_{1}, v_{2} of V such that v_{⟨⟨W_{1},W_{2}⟩⟩} = ⟨⟨v_{1}, v_{2}⟩⟩.

(19) Let K be a field, V be a vector space over K, v be a vector of V, X be a subspace of V, y be a vector of X + Lin({v}), and W be a subspace of X + Lin({v}). Suppose v = y and X = W and v ∉ X. Let w be a vector of X + Lin({v}). Then there exists a vector x of X and there exists an element r of the carrier of K such that w_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨x, r · v⟩⟩.

(20) Let K be a field, V be a vector space over K, v be a vector of V, X be a subspace of V, y be a vector of X + Lin({v}), and W be a subspace of X+Lin({v}). Suppose v = y and X = W and v ∉ X. Let w_{1}, w_{2} be vectors of X+Lin({v}), x_{1}, x_{2} be vectors of X, and r_{1}, r_{2} be elements of the carrier of K. If (w_{1})_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨x_{1}, r_{1} · v⟩⟩ and (w_{2})_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨x_{2}, r_{2} · v⟩⟩, then (w_{1} + w_{2})_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨x_{1} + x_{2}, (r_{1} + r_{2}) · v⟩⟩.

(21) Let K be a field, V be a vector space over K, v be a vector of V, X be a subspace of V, y be a vector of X + Lin({v}), and W be a subspace of X + Lin({v}). Suppose v = y and X = W and v ∉ X. Let w be a vector of X + Lin({v}), x be a vector of X, and t, r be elements of the carrier of K. If w_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨x, r · v⟩⟩, then (t · w)_{⟨⟨W,Lin({y})⟩⟩} = ⟨⟨t · x, t · r · v⟩⟩.

2. Quotient Vector Space for Non-Commutative Double Loop

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, and let W be a subspace of V. The functor CosetSet(V, W) yielding a non empty family of subsets of the carrier of V is defined as follows:

(Def. 2) CosetSet(V, W) = {A : A ranges over cosets of W }.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, and let W be a subspace of V. The functor addCoset(V, W) yields a binary operation on CosetSet(V, W) and is defined by:

(Def. 3) For all elements A, B of CosetSet(V, W) and for all vectors a, b of V such

that A = a+W and B = b+W holds (addCoset(V, W))(A, B) = a+b+W. Let K be an add-associative right zeroed right complementable Abelian as-

sociative left unital distributive non empty double loop structure, let V be a vector space over K, and let W be a subspace of V. The functor zeroCoset(V, W) yielding an element of CosetSet(V, W) is defined as follows:

(Def. 4) zeroCoset(V, W) = the carrier of W.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, and let W be a subspace of V. The functor lmultCoset(V, W) yields a function from [: the carrier of K, CosetSet(V, W) :] into CosetSet(V, W) and is defined by the condition (Def. 5).

(Def. 5) Let z be an element of the carrier of K, A be an element of CosetSet(V, W), and a be a vector of V. If A = a + W, then (lmultCoset(V, W))(z, A) = z · a + W.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, and let W be a subspace of V. The functor ^{V} /_{W} yielding a strict Abelian add-associative right zeroed right complementable vector space-like non empty vector space structure over K is defined by the conditions (Def. 6).

(Def. 6)(i) The carrier of ^{V} /_{W} = CosetSet(V, W), (ii) the addition of ^{V} /_{W} = addCoset(V, W), (iii) the zero of ^{V} /_{W} = zeroCoset(V, W), and (iv) the left multiplication of ^{V} /_{W} = lmultCoset(V, W). The following propositions are true:

(22) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, and W be a subspace of V. Then zeroCoset(V, W) = 0_{V} + W and 0_{^{V} /_{W}} = zeroCoset(V, W).

(23) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, W be a subspace of V, and w be a vector of V /_{W}. Then w is a coset of W and there exists a vector v of V such that w = v + W.

(24) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, W be a subspace of V, and v be a vector of V. Then v + W is a coset of W and v + W is a vector of ^{V} /_{W}.

(25) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, and W be a subspace of V. Then every coset of W is a vector of ^{V} /_{W}.

(26) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, W be a subspace of V, A be a vector of ^{V} /_{W}, v be a vector of V, and a be a scalar of V. If A = v + W, then a · A = a · v + W.

(27) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, W be a subspace of V, A_{1}, A_{2} be vectors of ^{V} /_{W}, and v_{1}, v_{2} be vectors of V. If A_{1} = v_{1} + W and A_{2} = v_{2} + W, then A_{1} + A_{2} = v_{1} + v_{2} + W.

3. Auxiliary Facts about Functionals

Next we state the proposition

(28) Let K be a field, V be a vector space over K, X be a subspace of V, f_{1} be a linear functional in X, v be a vector of V, and y be a vector of X +Lin({v}). Suppose v = y and v ∉ X. Let r be an element of the carrier of K. Then there exists a linear functional p_{1} in X + Lin({v}) such that p_{1}↾the carrier of X = f_{1} and p_{1}(y) = r.

Let K be a right zeroed non empty loop structure and let V be a non empty vector space structure over K. One can verify that there exists a functional in V which is additive and 0-preserving.

Let K be an add-associative right zeroed right complementable non empty double loop structure and let V be a right zeroed non empty vector space structure over K. Observe that every functional in V which is additive is also 0preserving.

Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure and let V be an addassociative right zeroed right complementable vector space-like non empty vector space structure over K. One can verify that every functional in V which is homogeneous is also 0-preserving.

Let K be a non empty zero structure and let V be a non empty vector space structure over K. One can check that 0FunctionalV is constant.

Let K be a non empty zero structure and let V be a non empty vector space structure over K. Note that there exists a functional in V which is constant.

Let K be an add-associative right zeroed right complementable non empty double loop structure, let V be a right zeroed non empty vector space structure over K, and let f be a 0-preserving functional in V. Let us observe that f is constant if and only if:

(Def. 7) f = 0FunctionalV.

Let K be an add-associative right zeroed right complementable non empty double loop structure and let V be a right zeroed non empty vector space structure over K. Note that there exists a functional in V which is constant, additive, and 0-preserving.

Let K be a non empty 1-sorted structure and let V be a non empty vector space structure over K. One can check that every functional in V which is non constant is also non trivial.

Let K be a field and let V be a non trivial vector space over K. Observe that there exists a functional in V which is additive, homogeneous, non constant, and non trivial.

Let K be a field and let V be a non trivial vector space over K. One can check that every functional in V which is trivial is also constant.

Let K be a field, let V be a non trivial vector space over K, let v be a vector of V, and let W be a linear complement of Lin({v}). Let us assume that v ≠ 0_{V} . The functor coeffFunctional(v, W) yielding a non constant non trivial linear functional in V is defined as follows:

(Def. 8) (coeffFunctional(v, W))(v) = 1_{K} and coeffFunctional(v, W)↾the carrier

of W = 0FunctionalW. We now state several propositions:

(29) Let K be a field, V be a non trivial vector space over K, and f be a non constant 0-preserving functional in V. Then there exists a vector v of V such that v ≠ 0_{V} and f(v) ≠ 0_{K}.

(30) Let K be a field, V be a non trivial vector space over K, v be a vector of V, a be a scalar of V, and W be a linear complement of Lin({v}). If v ≠ 0_{V} , then (coeffFunctional(v, W))(a · v) = a.

(31) Let K be a field, V be a non trivial vector space over K, v, w be vectors of V, and W be a linear complement of Lin({v}). If v ≠ 0_{V} and w ∈ W, then (coeffFunctional(v, W))(w) = 0_{K}.

(32) Let K be a field, V be a non trivial vector space over K, v, w be vectors of V, a be a scalar of V, and W be a linear complement of Lin({v}). If v ≠ 0_{V} and w ∈ W, then (coeffFunctional(v, W))(a · v + w) = a.

(33) Let K be a non empty loop structure, V be a non empty vector space structure over K, f, g be functionals in V, and v be a vector of V. Then

(f − g)(v) = f(v) − g(v). Let K be a field and let V be a non trivial vector space over K. Note that

\overline{V} is non trivial.

4. Kernel of Additive Functional. Linear Functionals in Quotient Vector Spaces

Let K be a non empty zero structure, let V be a non empty vector space structure over K, and let f be a functional in V. The functor kerf yields a subset of the carrier of V and is defined by:

(Def. 9) kerf = {v; v ranges over vectors of V: f(v) = 0_{K}}.

Let K be a right zeroed non empty loop structure, let V be a non empty vector space structure over K, and let f be a 0-preserving functional in V. One can check that kerf is non empty.

One can prove the following proposition

(34) Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, V be an add-associative right zeroed right complementable vector space-like non empty vector space structure over K, and f be a linear functional in V. Then kerf is linearly closed.

Let K be a non empty zero structure, let V be a non empty vector space structure over K, and let f be a functional in V. We say that f is degenerated if and only if:

(Def. 10) kerf ≠ {0_{V} }.

Let K be a non degenerated non empty double loop structure and let V be a non trivial non empty vector space structure over K. One can check that every functional in V which is non degenerated and 0-preserving is also non constant.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, and let f be a linear functional in V. The functor Kerf yields a strict non empty subspace of V and is defined as follows:

(Def. 11) The carrier of Kerf = kerf.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, let W be a subspace of V, and let f be an additive functional in V. Let us assume that the carrier of W ⊆ kerf. The functor ^{f}/_{W} yielding an additive functional in ^{V} /_{W} is defined by:

(Def. 12) For every vector A of ^{V} /_{W} and for every vector v of V such that A =

v + W holds (^{f}/_{W})(A) = f(v). One can prove the following proposition

quotient vector spaces and functionals 67

(35) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, W be a subspace of V, and f be a linear functional in V. If the carrier of W ⊆ kerf, then ^{f}/_{W} is homogeneous.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, and let f be a linear functional in V. The functor CQFunctionalf yielding a linear functional in ^{V} /_{Ker f} is defined as follows:

(Def. 13) CQFunctionalf = ^{f}/_{Ker f}. One can prove the following proposition

(36) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, f be a linear functional in V, A be a vector of^{V} /_{Ker f}, and v be a vector of V. If A = v + Kerf, then CQFunctionalf(A) = f(v).

Let K be a field, let V be a non trivial vector space over K, and let f be a non constant linear functional in V. Observe that CQFunctionalf is non constant.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, and let f be a linear functional in V. One can verify that CQFunctionalf is non degenerated.

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Bilinear Functionals in Vector Spaces^{1}

Jarosław Kotowicz University of Białystok

Summary. The main goal of the article is the presentation of the theory of bilinear functionals in vector spaces. It introduces standard operations on bilinear functionals and proves their classical properties. It is shown that quotient functionals are non-degenerate on the left and the right. In the case of symmetric and alternating bilinear functionals it is shown that the left and right kernels are equal.

MML Identifier: BILINEAR.

The papers [13], [6], [17], [12], [4], [18], [11], [2], [16], [3], [9], [19], [5], [7], [1], [15], [14], [10], and [8] provide the notation and terminology for this paper.

1. Two Form on Vector Spaces and Operations on Them

Let K be a 1-sorted structure and let V, W be vector space structures over K.

(Def. 1) A function from [: the carrier of V, the carrier of W :] into the carrier of

K is said to be a form of V, W. Let K be a non empty zero structure and let V, W be vector space structures

over K. The functor NulForm(V, W) yielding a form of V, W is defined by: (Def. 2) NulForm(V, W) = [: the carrier of V, the carrier of W :] ⟼→ 0_{K}.

Let K be a non empty loop structure, let V, W be non empty vector space structures over K, and let f, g be forms of V, W. The functor f + g yields a form of V, W and is defined as follows:

(Def. 3) For every vector v of V and for every vector w of W holds (f + g)(⟨⟨v, w⟩⟩) = f(⟨⟨v, w⟩⟩) + g(⟨⟨v, w⟩⟩).

1This work has been partially supported by TRIAL-SOLUTION grant IST-2001-35447 and SPUB-M grant of KBN.

Let K be a non empty groupoid, let V, W be non empty vector space structures over K, let f be a form of V, W, and let a be an element of the carrier of K. The functor a · f yields a form of V, W and is defined by:

(Def. 4) For every vector v of V and for every vector w of W holds (a · f)(⟨⟨v,

w⟩⟩) = a · f(⟨⟨v, w⟩⟩). Let K be a non empty loop structure, let V, W be non empty vector space

structures over K, and let f be a form of V, W. The functor − f yielding a form of V, W is defined as follows:

(Def. 5) For every vector v of V and for every vector w of W holds (− f)(⟨⟨v,

w⟩⟩) = − f(⟨⟨v, w⟩⟩). Let K be an add-associative right zeroed right complementable left distribu-

tive left unital non empty double loop structure, let V, W be non empty vector space structures over K, and let f be a form of V, W. Then − f is a form of V, W and it can be characterized by the condition:

(Def. 6) − f = (− 1_{K}) · f.

Let K be a non empty loop structure, let V, W be non empty vector space structures over K, and let f, g be forms of V, W. The functor f − g yields a form of V, W and is defined by:

(Def. 7) f − g = f + − g.

Let K be a non empty loop structure, let V, W be non empty vector space structures over K, and let f, g be forms of V, W. Then f − g is a form of V, W and it can be characterized by the condition:

(Def. 8) For every vector v of V and for every vector w of W holds (f − g)(⟨⟨v,

w⟩⟩) = f(⟨⟨v, w⟩⟩) − g(⟨⟨v, w⟩⟩). Let K be an Abelian non empty loop structure, let V, W be non empty

vector space structures over K, and let f, g be forms of V, W. Let us notice that the functor f + g is commutative.

Next we state several propositions:

(1) Let K be a non empty zero structure, V, W be non empty vector space structures over K, v be a vector of V, and w be a vector of W. Then (NulForm(V, W))(⟨⟨v, w⟩⟩) = 0_{K}.

(2) Let K be a right zeroed non empty loop structure, V, W be non empty vector space structures over K, and f be a form of V, W. Then f + NulForm(V, W) = f.

(3) Let K be an add-associative non empty loop structure, V, W be non empty vector space structures over K, and f, g, h be forms of V, W. Then (f + g) + h = f + (g + h).

(4) Let K be an add-associative right zeroed right complementable non empty loop structure, V, W be non empty vector space structures over K, and f be a form of V, W. Then f − f = NulForm(V, W).

(5) Let K be a right distributive non empty double loop structure, V, W be non empty vector space structures over K, a be an element of the carrier of K, and f, g be forms of V, W. Then a · (f + g) = a · f + a · g.

(6) Let K be a left distributive non empty double loop structure, V, W be non empty vector space structures over K, a, b be elements of the carrier of K, and f be a form of V, W. Then (a + b) · f = a · f + b · f.

(7) Let K be an associative non empty double loop structure, V, W be non empty vector space structures over K, a, b be elements of the carrier of K, and f be a form of V, W. Then (a · b) · f = a · (b · f).

(8) Let K be a left unital non empty double loop structure, V, W be non empty vector space structures over K, and f be a form of V, W. Then 1_{K} · f = f.

2. Functional Generated by Two Form when the One of Arguments is Fixed

Let K be a non empty 1-sorted structure, let V, W be non empty vector space structures over K, let f be a form of V, W, and let v be a vector of V. The functor f(v, ·) yielding a functional in W is defined as follows:

(Def. 9) f(v, ·) = (curryf)(v).

Let K be a non empty 1-sorted structure, let V, W be non empty vector space structures over K, let f be a form of V, W, and let w be a vector of W. The functor f(·, w) yields a functional in V and is defined by:

(Def. 10) f(·, w) = (curry′f)(w). The following propositions are true:

(9) Let K be a non empty 1-sorted structure, V, W be non empty vector space structures over K, f be a form of V, W, and v be a vector of V. Then domf(v, ·) = the carrier of W and rngf(v, ·) ⊆ the carrier of K and for every vector w of W holds (f(v, ·))(w) = f(⟨⟨v, w⟩⟩).

(10) Let K be a non empty 1-sorted structure, V, W be non empty vector space structures over K, f be a form of V, W, and w be a vector of W. Then domf(·, w) = the carrier of V and rngf(·, w) ⊆ the carrier of K and for every vector v of V holds (f(·, w))(v) = f(⟨⟨v, w⟩⟩).

(11) Let K be a non empty zero structure, V, W be non empty vector space structures over K, f be a form of V, W, and v be a vector of V. Then NulForm(V, W)(v, ·) = 0FunctionalW.

(12) Let K be a non empty zero structure, V, W be non empty vector space structures over K, f be a form of V, W, and w be a vector of W. Then NulForm(V, W)(·, w) = 0FunctionalV.

(13) Let K be a non empty loop structure, V, W be non empty vector space structures over K, f, g be forms of V, W, and w be a vector of W. Then (f + g)(·, w) = f(·, w) + g(·, w).

(14) Let K be a non empty loop structure, V, W be non empty vector space structures over K, f, g be forms of V, W, and v be a vector of V. Then (f + g)(v, ·) = f(v, ·) + g(v, ·).

(15) Let K be a non empty double loop structure, V, W be non empty vector space structures over K, f be a form of V, W, a be an element of the carrier of K, and w be a vector of W. Then (a · f)(·, w) = a · f(·, w).

(16) Let K be a non empty double loop structure, V, W be non empty vector space structures over K, f be a form of V, W, a be an element of the carrier of K, and v be a vector of V. Then (a · f)(v, ·) = a · f(v, ·).

(17) Let K be a non empty loop structure, V, W be non empty vector space structures over K, f be a form of V, W, and w be a vector of W. Then (− f)(·, w) = − f(·, w).

(18) Let K be a non empty loop structure, V, W be non empty vector space structures over K, f be a form of V, W, and v be a vector of V. Then (− f)(v, ·) = − f(v, ·).

(19) Let K be a non empty loop structure, V, W be non empty vector space structures over K, f, g be forms of V, W, and w be a vector of W. Then (f − g)(·, w) = f(·, w) − g(·, w).

(20) Let K be a non empty loop structure, V, W be non empty vector space structures over K, f, g be forms of V, W, and v be a vector of V. Then (f − g)(v, ·) = f(v, ·) − g(v, ·).

3. Two Form Generated by Functionals

Let K be a non empty groupoid, let V, W be non empty vector space structures over K, let f be a functional in V, and let g be a functional in W. The functor f ⊗ g yields a form of V, W and is defined as follows:

(Def. 11) For every vector v of V and for every vector w of W holds f ⊗ g(⟨⟨v,

w⟩⟩) = f(v) · g(w). One can prove the following propositions:

(21) Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, V, W be non empty vector space structures over K, f be a functional in V, v be a vector of V, and w be a vector of W. Then f ⊗ (0FunctionalW)(⟨⟨v, w⟩⟩) = 0_{K}.

(22) Let K be an add-associative right zeroed right complementable left distributive non empty double loop structure, V, W be non empty vector space structures over K, g be a functional in W, v be a vector of V, and w be a vector of W. Then (0FunctionalV) ⊗ g(⟨⟨v, w⟩⟩) = 0_{K}.

(23) Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, V, W be non empty vector space structures over K, and f be a functional in V. Then f ⊗ (0FunctionalW) = NulForm(V, W).

(24) Let K be an add-associative right zeroed right complementable left distributive non empty double loop structure, V, W be non empty vector space structures over K, and g be a functional in W. Then (0FunctionalV) ⊗ g = NulForm(V, W).

(25) Let K be a non empty groupoid, V, W be non empty vector space structures over K, f be a functional in V, g be a functional in W, and v be a vector of V. Then (f ⊗ g)(v, ·) = f(v) · g.

(26) Let K be a commutative non empty groupoid, V, W be non empty vector space structures over K, f be a functional in V, g be a functional in W, and w be a vector of W. Then (f ⊗ g)(·, w) = g(w) · f.

4. Bilinear Forms and their Properties

Let K be a non empty loop structure, let V, W be non empty vector space structures over K, and let f be a form of V, W. We say that f is additive wrt. second argument if and only if:

(Def. 12) For every vector v of V holds f(v, ·) is additive.

We say that f is additive wrt. first argument if and only if: (Def. 13) For every vector w of W holds f(·, w) is additive.

Let K be a non empty groupoid, let V, W be non empty vector space structures over K, and let f be a form of V, W. We say that f is homogeneous wrt. second argument if and only if:

(Def. 14) For every vector v of V holds f(v, ·) is homogeneous.

We say that f is homogeneous wrt. first argument if and only if: (Def. 15) For every vector w of W holds f(·, w) is homogeneous.

Let K be a right zeroed non empty loop structure and let V, W be non empty vector space structures over K. Note that NulForm(V, W) is additive wrt. second argument and NulForm(V, W) is additive wrt. first argument.

Let K be a right zeroed non empty loop structure and let V, W be non empty vector space structures over K. Note that there exists a form of V, W which is additive wrt. second argument and additive wrt. first argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure and let V, W be non empty vector space structures over K. Observe that NulForm(V, W) is homogeneous wrt. second argument and NulForm(V, W) is homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure and let V, W be non empty vector space structures over K. One can verify that there exists a form of V, W which is additive wrt. second argument, homogeneous wrt. second argument, additive wrt. first argument, and homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure and let V, W be non empty vector space structures over K. A bilinear form of V, W is an additive wrt. first argument homogeneous wrt. first argument additive wrt. second argument homogeneous wrt. second argument form of V, W.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be an additive wrt. second argument form of V, W, and let v be a vector of V. Note that f(v, ·) is additive.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be an additive wrt. first argument form of V, W, and let w be a vector of W. One can check that f(·, w) is additive.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be a homogeneous wrt. second argument form of V, W, and let v be a vector of V. Note that f(v, ·) is homogeneous.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be a homogeneous wrt. first argument form of V, W, and let w be a vector of W. One can verify that f(·, w) is homogeneous.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be a functional in V, and let g be an additive functional in W. One can check that f ⊗ g is additive wrt. second argument.

Let K be an add-associative right zeroed right complementable commutative right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be an additive functional in V, and let g be a functional in W. Note that f ⊗ g is additive wrt. first argument.

Let K be an add-associative right zeroed right complementable commutative associative right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be a functional in V, and let g be a homogeneous functional in W. Note that f ⊗ g is homogeneous wrt. second argument.

Let K be an add-associative right zeroed right complementable commutative associative right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be a homogeneous functional in V, and let g be a functional in W. Note that f ⊗ g is homogeneous wrt. first argument.

Let K be a non degenerated non empty double loop structure, let V be a non trivial non empty vector space structure over K, let W be a non empty vector space structure over K, let f be a functional in V, and let g be a functional in W. One can verify that f ⊗ g is non trivial.

Let K be a non degenerated non empty double loop structure, let V be a non empty vector space structure over K, let W be a non trivial non empty vector space structure over K, let f be a functional in V, and let g be a functional in W. One can verify that f ⊗ g is non trivial.

Let K be a field, let V, W be non trivial vector spaces over K, let f be a non constant 0-preserving functional in V, and let g be a non constant 0-preserving functional in W. Observe that f ⊗ g is non constant.

Let K be a field and let V, W be non trivial vector spaces over K. Observe that there exists a form of V, W which is non trivial, non constant, additive wrt. second argument, homogeneous wrt. second argument, additive wrt. first argument, and homogeneous wrt. first argument.

Let K be an Abelian add-associative right zeroed non empty loop structure, let V, W be non empty vector space structures over K, and let f, g be additive wrt. first argument forms of V, W. Observe that f + g is additive wrt. first argument.

Let K be an Abelian add-associative right zeroed non empty loop structure, let V, W be non empty vector space structures over K, and let f, g be additive wrt. second argument forms of V, W. Observe that f +g is additive wrt. second argument.

Let K be a right distributive right zeroed non empty double loop structure, let V, W be non empty vector space structures over K, let f be an additive wrt. first argument form of V, W, and let a be an element of the carrier of K. Observe that a · f is additive wrt. first argument.

Let K be a right distributive right zeroed non empty double loop structure, let V, W be non empty vector space structures over K, let f be an additive wrt. second argument form of V, W, and let a be an element of the carrier of K. Observe that a · f is additive wrt. second argument.

Let K be an Abelian add-associative right zeroed right complementable non empty loop structure, let V, W be non empty vector space structures over K, and let f be an additive wrt. first argument form of V, W. One can verify that − f is additive wrt. first argument.

Let K be an Abelian add-associative right zeroed right complementable non empty loop structure, let V, W be non empty vector space structures over K, and let f be an additive wrt. second argument form of V, W. One can check that − f is additive wrt. second argument.

Let K be an Abelian add-associative right zeroed right complementable non empty loop structure, let V, W be non empty vector space structures over K, and let f, g be additive wrt. first argument forms of V, W. Observe that f − g is additive wrt. first argument.

Let K be an Abelian add-associative right zeroed right complementable non empty loop structure, let V, W be non empty vector space structures over K, and let f, g be additive wrt. second argument forms of V, W. Note that f − g is additive wrt. second argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, and let f, g be homogeneous wrt. first argument forms of V, W. One can verify that f + g is homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, and let f, g be homogeneous wrt. second argument forms of V, W. Note that f + g is homogeneous wrt. second argument.

Let K be an add-associative right zeroed right complementable associative commutative right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be a homogeneous wrt. first argument form of V, W, and let a be an element of the carrier of K. One can check that a · f is homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable associative commutative right distributive non empty double loop structure, let V, W be non empty vector space structures over K, let f be a homogeneous wrt. second argument form of V, W, and let a be an element of the carrier of K. One can check that a · f is homogeneous wrt. second argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, and let f be a homogeneous wrt. first argument form of V, W. One can verify that − f is homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, and let f be a homogeneous wrt. second argument form of V, W. Note that − f is homogeneous wrt. second argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, and let f, g be homogeneous wrt. first argument forms of V, W. One can check that f − g is homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V, W be non empty vector space structures over K, and let f, g be homogeneous wrt. second argument forms of V, W. One can check that f − g is homogeneous wrt. second argument.

We now state a number of propositions:

(27) Let K be a non empty loop structure, V, W be non empty vector space structures over K, v, u be vectors of V, w be a vector of W, and f be a form of V, W. If f is additive wrt. first argument, then f(⟨⟨v + u, w⟩⟩) = f(⟨⟨v, w⟩⟩) + f(⟨⟨u, w⟩⟩).

(28) Let K be a non empty loop structure, V, W be non empty vector space structures over K, v be a vector of V, u, w be vectors of W, and f be a form of V, W. If f is additive wrt. second argument, then f(⟨⟨v, u+w⟩⟩) = f(⟨⟨v, u⟩⟩) + f(⟨⟨v, w⟩⟩).

(29) Let K be a right zeroed non empty loop structure, V, W be non empty vector space structures over K, v, u be vectors of V, w, t be vectors of W, and f be an additive wrt. first argument additive wrt. second argument form of V, W. Then f(⟨⟨v + u, w + t⟩⟩) = f(⟨⟨v, w⟩⟩) + f(⟨⟨v, t⟩⟩) + (f(⟨⟨u, w⟩⟩) + f(⟨⟨u, t⟩⟩)).

(30) Let K be an add-associative right zeroed right complementable non empty double loop structure, V, W be right zeroed non empty vector space structures over K, f be an additive wrt. second argument form of V, W, and v be a vector of V. Then f(⟨⟨v, 0_{W} ⟩⟩) = 0_{K}.

(31) Let K be an add-associative right zeroed right complementable non empty double loop structure, V, W be right zeroed non empty vector space structures over K, f be an additive wrt. first argument form of V, W, and w be a vector of W. Then f(⟨⟨0_{V} , w⟩⟩) = 0_{K}.

(32) Let K be a non empty groupoid, V, W be non empty vector space structures over K, v be a vector of V, w be a vector of W, a be an element of the carrier of K, and f be a form of V, W. If f is homogeneous wrt. first argument, then f(⟨⟨a · v, w⟩⟩) = a · f(⟨⟨v, w⟩⟩).

(33) Let K be a non empty groupoid, V, W be non empty vector space structures over K, v be a vector of V, w be a vector of W, a be an element of the carrier of K, and f be a form of V, W. If f is homogeneous wrt. second argument, then f(⟨⟨v, a · w⟩⟩) = a · f(⟨⟨v, w⟩⟩).

(34) Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, V, W be add-associative right zeroed right complementable vector space-like non empty vector space structures over K, f be a homogeneous wrt. first argument form of V, W, and w be a vector of W. Then f(⟨⟨0_{V} , w⟩⟩) = 0_{K}.

(35) Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, V, W be add-associative right zeroed right complementable vector space-like non empty vector space structures over K, f be a homogeneous wrt. second

argument form of V, W, and v be a vector of V. Then f(⟨⟨v, 0_{W} ⟩⟩) = 0_{K}.

(36) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, v, u be vectors of V, w be a vector of W, and f be an additive wrt. first argument homogeneous wrt. first argument form of V, W. Then f(⟨⟨v − u, w⟩⟩) = f(⟨⟨v, w⟩⟩) − f(⟨⟨u, w⟩⟩).

(37) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, v be a vector of V, w, t be vectors of W, and f be an additive wrt. second argument homogeneous wrt. second argument form of V, W. Then f(⟨⟨v, w − t⟩⟩) = f(⟨⟨v, w⟩⟩) − f(⟨⟨v, t⟩⟩).

(38) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, v, u be vectors of V, w, t be vectors of W, and f be a bilinear form of V, W. Then f(⟨⟨v − u, w − t⟩⟩) = f(⟨⟨v, w⟩⟩) − f(⟨⟨v, t⟩⟩) − (f(⟨⟨u, w⟩⟩) − f(⟨⟨u, t⟩⟩)).

(39) Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, V, W be add-associative right zeroed right complementable vector space-like non empty vector space structures over K, v, u be vectors of V, w, t be vectors of W, a, b be elements of the carrier of K, and f be a bilinear form of V, W. Then f(⟨⟨v + a · u, w + b · t⟩⟩) = f(⟨⟨v, w⟩⟩) + b · f(⟨⟨v, t⟩⟩) + (a · f(⟨⟨u, w⟩⟩) + a · (b · f(⟨⟨u, t⟩⟩))).

(40) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, v, u be vectors of V, w, t be vectors of W, a, b be elements of the carrier of K, and f be a bilinear form of V, W. Then f(⟨⟨v −a·u, w−b·t⟩⟩) = f(⟨⟨v, w⟩⟩)−b·f(⟨⟨v, t⟩⟩)−(a·f(⟨⟨u, w⟩⟩)−a·(b·f(⟨⟨u, t⟩⟩))).

(41) Let K be an add-associative right zeroed right complementable non empty double loop structure, V, W be right zeroed non empty vector space structures over K, and f be a form of V, W. Suppose f is additive wrt. second argument and additive wrt. first argument. Then f is constant if and only if for every vector v of V and for every vector w of W holds f(⟨⟨v, w⟩⟩) = 0_{K}.

5. Left and Right Kernel of Form. Kernel of “Diagonal”

Let K be a zero structure, let V, W be non empty vector space structures over K, and let f be a form of V, W. The functor leftkerf yields a subset of the carrier of V and is defined as follows:

(Def. 16) leftkerf = {v; v ranges over vectors of V: ⋀_{w: vector of W} f(⟨⟨v, w⟩⟩) =

0_{K}}. Let K be a zero structure, let V, W be non empty vector space structures

over K, and let f be a form of V, W. The functor rightkerf yielding a subset of the carrier of W is defined by:

(Def. 17) rightkerf = {w; w ranges over vectors of W: ⋀_{v: vector of V} f(⟨⟨v, w⟩⟩) =

0_{K}}. Let K be a zero structure, let V be a non empty vector space structure over

K, and let f be a form of V, V. The functor diagkerf yielding a subset of the carrier of V is defined by:

(Def. 18) diagkerf = {v; v ranges over vectors of V: f(⟨⟨v, v⟩⟩) = 0_{K}}.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V be a right zeroed non empty vector space structure over K, let W be a non empty vector space structure over K, and let f be an additive wrt. first argument form of V, W. Note that leftkerf is non empty.

Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, let V be an addassociative right zeroed right complementable vector space-like non empty vector space structure over K, let W be a non empty vector space structure over K, and let f be a homogeneous wrt. first argument form of V, W. Observe that leftkerf is non empty.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V be a non empty vector space structure over K, let W be a right zeroed non empty vector space structure over K, and let f be an additive wrt. second argument form of V, W. One can verify that rightkerf is non empty.

Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, let V be a non empty vector space structure over K, let W be an add-associative right zeroed right complementable vector space-like non empty vector space structure over K, and let f be a homogeneous wrt. second argument form of V, W. One can check that rightkerf is non empty.

Let K be an add-associative right zeroed right complementable non empty double loop structure, let V be a right zeroed non empty vector space structure over K, and let f be an additive wrt. second argument form of V, V. Note that diagkerf is non empty.

Let K be an add-associative right zeroed right complementable non empty double loop structure, let V be a right zeroed non empty vector space structure over K, and let f be an additive wrt. first argument form of V, V. Note that diagkerf is non empty.

Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, let V be an addassociative right zeroed right complementable vector space-like non empty vector space structure over K, and let f be a homogeneous wrt. second argument form of V, V. One can check that diagkerf is non empty.

Let K be an add-associative right zeroed right complementable associative left unital distributive non empty double loop structure, let V be an addassociative right zeroed right complementable vector space-like non empty vector space structure over K, and let f be a homogeneous wrt. first argument form of V, V. One can check that diagkerf is non empty.

We now state three propositions:

(42) Let K be a zero structure, V be a non empty vector space structure over K, and f be a form of V, V. Then leftkerf ⊆ diagkerf and rightkerf ⊆ diagkerf.

(43) Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, V, W be non empty vector space structures over K, and f be an additive wrt. first argument homogeneous wrt. first argument form of V, W. Then leftkerf is linearly closed.

(44) Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, V, W be non empty vector space structures over K, and f be an additive wrt. second argument homogeneous wrt. second argument form of V, W. Then rightkerf is linearly closed.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, let W be a non empty vector space structure over K, and let f be an additive wrt. first argument homogeneous wrt. first argument form of V, W. The functor LKerf yielding a strict non empty subspace of V is defined by:

(Def. 19) The carrier of LKerf = leftkerf.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a non empty vector space structure over K, let W be a vector space over K, and let f be an additive wrt. second argument homogeneous wrt. second argument form of V, W. The functor RKerf yielding a strict non empty subspace of W is defined by:

(Def. 20) The carrier of RKerf = rightkerf.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, let W be a non empty vector space structure over K, and let f be an additive wrt. first argument homogeneous wrt. first argument form of V, W. The functor LQForm(f) yields an additive wrt. first argument homogeneous wrt. first argument form of ^{V} /_{LKer f}, W and is defined by the condition (Def. 21).

(Def. 21) Let A be a vector of ^{V} /_{LKer f}, w be a vector of W, and v be a vector of

V. If A = v + LKerf, then (LQForm(f))(⟨⟨A, w⟩⟩) = f(⟨⟨v, w⟩⟩). Let K be an add-associative right zeroed right complementable Abelian as-

sociative left unital distributive non empty double loop structure, let V be a non empty vector space structure over K, let W be a vector space over K, and let f be an additive wrt. second argument homogeneous wrt. second argument form of V, W. The functor RQForm(f) yielding an additive wrt. second argument homogeneous wrt. second argument form of V, ^{W} /_{RKer f} is defined by the condition (Def. 22).

(Def. 22) Let A be a vector of ^{W} /_{RKer f}, v be a vector of V, and w be a vector of

W. If A = w + RKerf, then (RQForm(f))(⟨⟨v, A⟩⟩) = f(⟨⟨v, w⟩⟩). Let K be an add-associative right zeroed right complementable Abelian as-

sociative left unital distributive non empty double loop structure, let V, W be vector spaces over K, and let f be a bilinear form of V, W. Note that LQForm(f) is additive wrt. second argument and homogeneous wrt. second argument and RQForm(f) is additive wrt. first argument and homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V, W be vector spaces over K, and let f be a bilinear form of V, W. The functor QForm(f) yields a bilinear form of ^{V} /_{LKer f}, ^{W} /_{RKer f} and is defined by the condition (Def. 23).

(Def. 23) Let A be a vector of ^{V} /_{LKer f}, B be a vector of ^{W} /_{RKer f}, v be a vector of V, and w be a vector of W. If A = v + LKerf and B = w + RKerf, then (QForm(f))(⟨⟨A, B⟩⟩) = f(⟨⟨v, w⟩⟩).

One can prove the following propositions:

(45) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a vector space over K, W be a non empty vector space structure over K, and f be an additive wrt. first argument homogeneous wrt. first argument form of V, W. Then rightkerf = rightker(LQForm(f)).

(46) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V be a non empty vector space structure over K, W be a vector space over K, and f be an additive wrt. second argument homogeneous wrt. second argument form of V, W. Then leftkerf = leftker(RQForm(f)).

(47) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, and f be a bilinear form of V, W. Then RKerf = RKer(LQForm(f)).

(48) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, and f be a bilinear form of V, W. Then LKerf = LKer(RQForm(f)).

(49) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, and f be a bilinear form of V, W. Then QForm(f) = RQForm(LQForm(f)) and QForm(f) = LQForm(RQForm(f)).

(50) Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, V, W be vector spaces over K, and f be a bilinear form of V, W. Then leftker(QForm(f)) = leftker(RQForm(LQForm(f))) and rightker(QForm(f)) = rightker(RQForm(LQForm(f))) and leftker(QForm (f)) = leftker(LQForm(RQForm(f))) and rightker(QForm(f)) = rightker(LQForm(RQForm(f))).

(51) Let K be an add-associative right zeroed right complementable distributive non empty double loop structure, V, W be non empty vector space structures over K, f be a functional in V, and g be a functional in W. Then kerf ⊆ leftker(f ⊗ g).

(52) Let K be an add-associative right zeroed right complementable associative commutative left unital field-like distributive non empty double loop structure, V, W be non empty vector space structures over K, f be a functional in V, and g be a functional in W. If g ≠ 0FunctionalW, then leftker(f ⊗ g) = kerf.

(53) Let K be an add-associative right zeroed right complementable distributive non empty double loop structure, V, W be non empty vector space structures over K, f be a functional in V, and g be a functional in W. Then kerg ⊆ rightker(f ⊗ g).

(54) Let K be an add-associative right zeroed right complementable associative commutative left unital field-like distributive non empty double loop structure, V, W be non empty vector space structures over K, f be a functional in V, and g be a functional in W. If f ≠ 0FunctionalV, then rightker(f ⊗ g) = kerg.

(55) Let K be an add-associative right zeroed right complementable commutative Abelian associative left unital distributive field-like non empty double loop structure, V be a vector space over K, W be a non empty vector space structure over K, f be a linear functional in V, and g be a functional in W. If g ≠ 0FunctionalW, then LKer(f ⊗ g) = Kerf and LQForm(f ⊗ g) = (CQFunctionalf) ⊗ g.

(56) Let K be an add-associative right zeroed right complementable commutative Abelian associative left unital distributive field-like non empty double loop structure, V be a non empty vector space structure over K, W be a vector space over K, f be a functional in V, and g be a linear functional in W. If f ≠ 0FunctionalV, then RKer(f ⊗ g) = Kerg and RQForm(f ⊗ g) = f ⊗ (CQFunctionalg).

(57) Let K be a field, V, W be non trivial vector spaces over K, f be a non constant linear functional in V, and g be a non constant linear functional in W. Then QForm(f ⊗ g) = (CQFunctionalf) ⊗ (CQFunctionalg).

Let K be a zero structure, let V, W be non empty vector space structures over K, and let f be a form of V, W. We say that f is degenerated on left if and only if:

(Def. 24) leftkerf ≠ {0_{V} }.

We say that f is degenerated on right if and only if: (Def. 25) rightkerf ≠ {0_{W} }.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a vector space over K, let W be a right zeroed non empty vector space structure over K, and let f be an additive wrt. first argument homogeneous wrt. first argument form of V, W. Note that LQForm(f) is non degenerated on left.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V be a right zeroed non empty vector space structure over K, let W be a vector space over K, and let f be an additive wrt. second argument homogeneous wrt. second argument form of V, W. Note that RQForm(f) is non degenerated on right.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V, W be vector spaces over K, and let f be a bilinear form of V, W. Observe that QForm(f) is non degenerated on left and non degenerated on right.

Let K be an add-associative right zeroed right complementable Abelian associative left unital distributive non empty double loop structure, let V, W be vector spaces over K, and let f be a bilinear form of V, W. One can verify that RQForm(LQForm(f)) is non degenerated on left and non degenerated on right and LQForm(RQForm(f)) is non degenerated on left and non degenerated on right.

Let K be a field, let V, W be non trivial vector spaces over K, and let f be a non constant bilinear form of V, W. Note that QForm(f) is non constant.

6. Bilinear Symmetric and Alternating Forms

Let K be a 1-sorted structure, let V be a vector space structure over K, and let f be a form of V, V. We say that f is symmetric if and only if:

(Def. 26) For all vectors v, w of V holds f(⟨⟨v, w⟩⟩) = f(⟨⟨w, v⟩⟩).

Let K be a zero structure, let V be a vector space structure over K, and let f be a form of V, V. We say that f is alternating if and only if:

(Def. 27) For every vector v of V holds f(⟨⟨v, v⟩⟩) = 0_{K}.

Let K be a non empty zero structure and let V be a non empty vector space structure over K. Observe that NulForm(V, V) is symmetric and NulForm(V, V) is alternating.

Let K be a non empty zero structure and let V be a non empty vector space structure over K. Observe that there exists a form of V, V which is symmetric and there exists a form of V, V which is alternating.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure and let V be a non empty vector space structure over K. Note that there exists a form of V, V which is symmetric, additive wrt. second argument, homogeneous wrt. second argument, additive wrt. first argument, and homogeneous wrt. first argument and there exists a form of V, V which is alternating, additive wrt. second argument, homogeneous wrt. second argument, additive wrt. first argument, and homogeneous wrt. first argument.

Let K be a field and let V be a non trivial vector space over K. Observe that there exists a form of V, V which is symmetric, non trivial, non constant, additive wrt. second argument, homogeneous wrt. second argument, additive wrt. first argument, and homogeneous wrt. first argument.

Let K be an add-associative right zeroed right complementable non empty loop structure and let V be a non empty vector space structure over K. Note that there exists a form of V, V which is alternating, additive wrt. second argument, and additive wrt. first argument.

Let K be a non empty loop structure, let V be a non empty vector space structure over K, and let f, g be symmetric forms of V, V. One can check that f + g is symmetric.

Let K be a non empty double loop structure, let V be a non empty vector space structure over K, let f be a symmetric form of V, V, and let a be an element of the carrier of K. One can check that a · f is symmetric.

Let K be a non empty loop structure, let V be a non empty vector space structure over K, and let f be a symmetric form of V, V. Note that − f is symmetric.

Let K be a non empty loop structure, let V be a non empty vector space structure over K, and let f, g be symmetric forms of V, V. Observe that f − g is symmetric.

Let K be a right zeroed non empty loop structure, let V be a non empty vector space structure over K, and let f, g be alternating forms of V, V. One can check that f + g is alternating.

Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, let V be a non empty vector space structure over K, let f be an alternating form of V, V, and let a be a scalar of K. One can verify that a · f is alternating.

Let K be an add-associative right zeroed right complementable non empty loop structure, let V be a non empty vector space structure over K, and let f be an alternating form of V, V. Note that − f is alternating.

Let K be an add-associative right zeroed right complementable non empty loop structure, let V be a non empty vector space structure over K, and let f, g be alternating forms of V, V. Observe that f − g is alternating.

One can prove the following two propositions:

(58) Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, V be a non empty vector space structure over K, and f be a symmetric bilinear form of V, V. Then leftkerf = rightkerf.

(59) Let K be an add-associative right zeroed right complementable non empty loop structure, V be a non empty vector space structure over K, f be an alternating additive wrt. first argument additive wrt. second argument form of V, V, and v, w be vectors of V. Then f(⟨⟨v, w⟩⟩) = − f(⟨⟨w, v⟩⟩).

Let K be a Fanoian field, let V be a non empty vector space structure over K, and let f be an additive wrt. first argument additive wrt. second argument form of V, V. Let us observe that f is alternating if and only if:

(Def. 28) For all vectors v, w of V holds f(⟨⟨v, w⟩⟩) = − f(⟨⟨w, v⟩⟩).

Next we state the proposition

(60) Let K be an add-associative right zeroed right complementable right distributive non empty double loop structure, V be a non empty vector space structure over K, and f be an alternating bilinear form of V, V. Then leftkerf = rightkerf.

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Hermitan Functionals.

Canonical Construction of Scalar Product in Quotient Vector Space^{1}

Jarosław Kotowicz University of Białystok

Summary. In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non-negative hermitan form. The proof of Schwarz inequality is based on [14]. The incorrect proof of this fact can be found in [11]. The construction of scalar product in quotient vector space from non-negative hermitan functions is the main result of the article.

MML Identifier: HERMITAN.

The notation and terminology used in this paper have been introduced in the following articles: [16], [5], [20], [6], [15], [3], [1], [19], [10], [21], [4], [17], [2], [7], [18], [12], [13], [9], and [8].

1. Auxiliary Facts about Complex Numbers

The following propositions are true:

(1) For every element a of ℂ such that a = \overline{a} holds ℑ(a) = 0.

(2) For every element a of ℂ such that a ≠ 0_{ℂ} holds |\frac{ℜ(a)}{|a|} + \frac{− ℑ(a)}{|a|} i| = 1 and

ℜ((\frac{ℜ(a)}{|a|} + \frac{− ℑ(a)}{|a|} i) · a) = |a| and ℑ((\frac{ℜ(a)}{|a|} + \frac{− ℑ(a)}{|a|} i) · a) = 0.

(3) For every element a of ℂ there exists an element b of ℂ such that |b| = 1 and ℜ(b · a) = |a| and ℑ(b · a) = 0.

(4) For every element a of ℂ holds a · \overline{a} = |a|^{2} + 0i.

1This work has been partially supported by TRIAL-SOLUTION grant IST-2001-35447 and SPUB-M grant of KBN.

(5) For every element a of the carrier of ℂ_{F} such that a = \overline{a} holds ℑ(a) = 0.

(6) \overline{i_{ℂ_{F}}} = (i)^{−1} .

(7) i_{ℂ_{F}} · \overline{i_{ℂ_{F}}} = 1_{ℂ_{F}}.

(8) Let a be an element of the carrier of ℂ_{F}. Suppose a ≠ 0_{ℂ_{F}}. Then |\frac{ℜ(a)}{|a|} +

\frac{− ℑ(a)}{|a|} i_{ℂ_{F}}| = 1 and ℜ((\frac{ℜ(a)}{|a|} + \frac{− ℑ(a)}{|a|} i_{ℂ_{F}}) · a) = |a| and ℑ((\frac{ℜ(a)}{|a|} + \frac{− ℑ(a)}{|a|} i_{ℂ_{F}}) · a) = 0.

(9) Let a be an element of the carrier of ℂ_{F}. Then there exists an element b of the carrier of ℂ_{F} such that |b| = 1 and ℜ(b · a) = |a| and ℑ(b · a) = 0.

(10) For all elements a, b of the carrier of ℂ_{F} holds ℜ(a − b) = ℜ(a) − ℜ(b) and ℑ(a − b) = ℑ(a) − ℑ(b).

(11) For all elements a, b of the carrier of ℂ_{F} such that ℑ(a) = 0 holds ℜ(a · b) = ℜ(a) · ℜ(b) and ℑ(a · b) = ℜ(a) · ℑ(b).

(12) For all elements a, b of the carrier of ℂ_{F} such that ℑ(a) = 0 and ℑ(b) = 0 holds ℑ(a · b) = 0.

(13) For every element a of the carrier of ℂ_{F} holds ℜ(a) = ℜ(\overline{a}).

(14) For every element a of the carrier of ℂ_{F} such that ℑ(a) = 0 holds a = \overline{a}.

(15) For all real numbers r, s holds (r + 0i_{ℂ_{F}}) · (s + 0i_{ℂ_{F}}) = r · s + 0i_{ℂ_{F}}.

(16) For every element a of the carrier of ℂ_{F} holds a · \overline{a} = |a|^{2} + 0i_{ℂ_{F}}.

(17) For every element a of the carrier of ℂ_{F} such that 0 ℜ(a) and ℑ(a) = 0 holds |a| = ℜ(a).

(18) For every element a of the carrier of ℂ_{F} holds ℜ(a) + ℜ(\overline{a}) = 2 · ℜ(a).

2. Antilinear Functionals in Complex Vector Spaces

Let V be a non empty vector space structure over ℂ_{F} and let f be a functional in V. We say that f is complex-homogeneous if and only if:

(Def. 1) For every vector v of V and for every scalar a of V holds f(a·v) = \overline{a}·f(v).

Let V be a non empty vector space structure over ℂ_{F}. Observe that 0FunctionalV is complex-homogeneous.

Let V be an add-associative right zeroed right complementable vector spacelike non empty vector space structure over ℂ_{F}. One can verify that every functional in V which is complex-homogeneous is also 0-preserving.

Let V be a non empty vector space structure over ℂ_{F}. One can check that there exists a functional in V which is additive, complex-homogeneous, and 0-preserving.

Let V be a non empty vector space structure over ℂ_{F}. An antilinear functional of V is an additive complex-homogeneous functional in V.

Let V be a non empty vector space structure over ℂ_{F} and let f, g be complexhomogeneous functionals in V. Observe that f + g is complex-homogeneous.

Let V be a non empty vector space structure over ℂ_{F} and let f be a complexhomogeneous functional in V. One can verify that − f is complex-homogeneous.

Let V be a non empty vector space structure over ℂ_{F}, let a be a scalar of V, and let f be a complex-homogeneous functional in V. One can verify that a · f is complex-homogeneous.

Let V be a non empty vector space structure over ℂ_{F} and let f, g be complex-homogeneous functionals in V. One can check that f − g is complexhomogeneous.

Let V be a non empty vector space structure over ℂ_{F} and let f be a functional in V. The functor \overline{f} yields a functional in V and is defined by:

(Def. 2) For every vector v of V holds \overline{f} (v) = \overline{f(v)}.

Let V be a non empty vector space structure over ℂ_{F} and let f be an additive functional in V. Note that \overline{f} is additive.

Let V be a non empty vector space structure over ℂ_{F} and let f be a homogeneous functional in V. Note that \overline{f} is complex-homogeneous.

Let V be a non empty vector space structure over ℂ_{F} and let f be a complexhomogeneous functional in V. Note that \overline{f} is homogeneous.

Let V be a non trivial vector space over ℂ_{F} and let f be a non constant functional in V. One can check that \overline{f} is non constant.

Let V be a non trivial vector space over ℂ_{F}. One can check that there exists a functional in V which is additive, complex-homogeneous, non constant, and non trivial.

The following propositions are true:

(19) For every non empty vector space structure V over ℂ_{F} and for every functional f in V holds \overline{\overline{f}} = f.

(20) For every non empty vector space structure V over ℂ_{F} holds \overline{0FunctionalV} = 0FunctionalV.

(21) For every non empty vector space structure V over ℂ_{F} and for all functionals f, g in V holds \overline{f + g} = \overline{f} + \overline{g}.

(22) For every non empty vector space structure V over ℂ_{F} and for every functional f in V holds \overline{− f} = − \overline{f} .

(23) Let V be a non empty vector space structure over ℂ_{F}, f be a functional in V, and a be a scalar of V. Then \overline{a · f} = \overline{a} · \overline{f} .

(24) For every non empty vector space structure V over ℂ_{F} and for all functionals f, g in V holds \overline{f − g} = \overline{f} − \overline{g}.

(25) Let V be a non empty vector space structure over ℂ_{F}, f be a functional in V, and v be a vector of V. Then f(v) = 0_{ℂ_{F}} if and only if \overline{f} (v) = 0_{ℂ_{F}}.

(26) For every non empty vector space structure V over ℂ_{F} and for every functional f in V holds kerf = ker \overline{f} .

(27) Let V be an add-associative right zeroed right complementable vector space-like non empty vector space structure over ℂ_{F} and f be an antilinear functional of V. Then kerf is linearly closed.

(28) Let V be a vector space over ℂ_{F}, W be a subspace of V, and f be an antilinear functional of V. If the carrier of W ⊆ ker \overline{f} , then ^{f}/_{W} is complex-homogeneous.

Let V be a vector space over ℂ_{F} and let f be an antilinear functional of V. The functor QcFunctionalf yields an antilinear functional of ^{V} /_{Ker \overline{f}} and is defined as follows:

(Def. 3) QcFunctionalf = ^{f}/_{Ker \overline{f}} .

We now state the proposition

(29) Let V be a vector space over ℂ_{F}, f be an antilinear functional of V, A be a vector of ^{V} /_{Ker \overline{f}}, and v be a vector of V. If A = v + Ker \overline{f} , then (QcFunctionalf)(A) = f(v).

Let V be a non trivial vector space over ℂ_{F} and let f be a non constant antilinear functional of V. One can check that QcFunctionalf is non constant.

Let V be a vector space over ℂ_{F} and let f be an antilinear functional of V. Observe that QcFunctionalf is non degenerated.

3. Sesquilinear Forms in Complex Vector Spaces

Let V, W be non empty vector space structures over ℂ_{F} and let f be a form of V, W. We say that f is complex-homogeneous wrt. second argument if and only if:

(Def. 4) For every vector v of V holds f(v, ·) is complex-homogeneous.

We now state the proposition

(30) Let V, W be non empty vector space structures over ℂ_{F}, v be a vector of V, w be a vector of W, a be an element of the carrier of ℂ_{F}, and f be a form of V, W. Suppose f is complex-homogeneous wrt. second argument. Then f(⟨⟨v, a · w⟩⟩) = \overline{a} · f(⟨⟨v, w⟩⟩).

Let V be a non empty vector space structure over ℂ_{F} and let f be a form of V, V. We say that f is hermitan if and only if:

(Def. 5) For all vectors v, u of V holds f(⟨⟨v, u⟩⟩) = \overline{f(⟨⟨u, v⟩⟩)}.

We say that f is diagonal real valued if and only if: (Def. 6) For every vector v of V holds ℑ(f(⟨⟨v, v⟩⟩)) = 0.

We say that f is diagonal plus-real valued if and only if: (Def. 7) For every vector v of V holds 0 ℜ(f(⟨⟨v, v⟩⟩)).

Let V, W be non empty vector space structures over ℂ_{F}. Observe that NulForm(V, W) is complex-homogeneous wrt. second argument.

Let V be a non empty vector space structure over ℂ_{F}. Observe that NulForm(V, V) is hermitan and NulForm(V, V) is diagonal plus-real valued.

Let V be a non empty vector space structure over ℂ_{F}. Observe that every form of V, V which is hermitan is also diagonal real valued.

Let V be a non empty vector space structure over ℂ_{F}. One can check that there exists a form of V, V which is diagonal plus-real valued, hermitan, diagonal real valued, additive wrt. first argument, homogeneous wrt. first argument, additive wrt. second argument, and complex-homogeneous wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F}. One can check that there exists a form of V, W which is additive wrt. first argument, homogeneous wrt. first argument, additive wrt. second argument, and complex-homogeneous wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F}. A sesquilinear form of V, W is an additive wrt. first argument homogeneous wrt. first argument additive wrt. second argument complex-homogeneous wrt. second argument form of V, W.

Let V be a non empty vector space structure over ℂ_{F}. One can check that every form of V, V which is hermitan and additive wrt. second argument is also additive wrt. first argument.

Let V be a non empty vector space structure over ℂ_{F}. Observe that every form of V, V which is hermitan and additive wrt. first argument is also additive wrt. second argument.

Let V be a non empty vector space structure over ℂ_{F}. Observe that every form of V, V which is hermitan and homogeneous wrt. first argument is also complex-homogeneous wrt. second argument.

Let V be a non empty vector space structure over ℂ_{F}. Note that every form of V, V which is hermitan and complex-homogeneous wrt. second argument is also homogeneous wrt. first argument.

Let V be a non empty vector space structure over ℂ_{F}. A hermitan form of V is a hermitan additive wrt. first argument homogeneous wrt. first argument form of V, V.

Let V, W be non empty vector space structures over ℂ_{F}, let f be a functional in V, and let g be a complex-homogeneous functional in W. Note that f ⊗ g is complex-homogeneous wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F}, let f be a complexhomogeneous wrt. second argument form of V, W, and let v be a vector of V. One can verify that f(v, ·) is complex-homogeneous.

Let V, W be non empty vector space structures over ℂ_{F} and let f, g be complex-homogeneous wrt. second argument forms of V, W. One can verify that f + g is complex-homogeneous wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F}, let f be a complexhomogeneous wrt. second argument form of V, W, and let a be a scalar of V. Observe that a · f is complex-homogeneous wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F} and let f be a complex-homogeneous wrt. second argument form of V, W. One can check that − f is complex-homogeneous wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F} and let f, g be complex-homogeneous wrt. second argument forms of V, W. Observe that f −g is complex-homogeneous wrt. second argument.

Let V, W be non trivial vector spaces over ℂ_{F}. Observe that there exists a form of V, W which is additive wrt. first argument, homogeneous wrt. first argument, additive wrt. second argument, complex-homogeneous wrt. second argument, non constant, and non trivial.

Let V, W be non empty vector space structures over ℂ_{F} and let f be a form of V, W. The functor \overline{f} yielding a form of V, W is defined by:

(Def. 8) For every vector v of V and for every vector w of W holds \overline{f} (⟨⟨v, w⟩⟩) =

\overline{f(⟨⟨v, w⟩⟩)}. Let V, W be non empty vector space structures over ℂ_{F} and let f be an

additive wrt. second argument form of V, W. Note that \overline{f} is additive wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F} and let f be an additive wrt. first argument form of V, W. Note that \overline{f} is additive wrt. first argument.

Let V, W be non empty vector space structures over ℂ_{F} and let f be a homogeneous wrt. second argument form of V, W. One can check that \overline{f} is complex-homogeneous wrt. second argument.

Let V, W be non empty vector space structures over ℂ_{F} and let f be a complex-homogeneous wrt. second argument form of V, W. Note that \overline{f} is homogeneous wrt. second argument.

Let V, W be non trivial vector spaces over ℂ_{F} and let f be a non constant form of V, W. One can verify that \overline{f} is non constant.

The following proposition is true

(31) Let V be a non empty vector space structure over ℂ_{F}, f be a functional

in V, and v be a vector of V. Then f ⊗ \overline{f} (⟨⟨v, v⟩⟩) = |f(v)|^{2} + 0i_{ℂ_{F}}. Let V be a non empty vector space structure over ℂ_{F} and let f be a functional

in V. One can verify that f ⊗ \overline{f} is diagonal plus-real valued, hermitan, and diagonal real valued.

Let V be a non trivial vector space over ℂ_{F}. Note that there exists a form of V, V which is diagonal plus-real valued, hermitan, diagonal real valued, additive wrt. first argument, homogeneous wrt. first argument, additive wrt. second argument, complex-homogeneous wrt. second argument, non constant, and non trivial.

We now state a number of propositions:

(32) For all non empty vector space structures V, W over ℂ_{F} and for every form f of V, W holds \overline{\overline{f}} = f.

(33) For all non empty vector space structures V, W over ℂ_{F} holds \overline{NulForm(V, W)} = NulForm(V, W).

(34) For all non empty vector space structures V, W over ℂ_{F} and for all forms f, g of V, W holds \overline{f + g} = \overline{f} + \overline{g}.

(35) For all non empty vector space structures V, W over ℂ_{F} and for every form f of V, W holds \overline{− f} = − \overline{f} .

(36) Let V, W be non empty vector space structures over ℂ_{F}, f be a form of V, W, and a be an element of ℂ_{F}. Then \overline{a · f} = \overline{a} · \overline{f} .

(37) For all non empty vector space structures V, W over ℂ_{F} and for all forms f, g of V, W holds \overline{f − g} = \overline{f} − \overline{g}.

(38) Let V, W be vector spaces over ℂ_{F}, v be a vector of V, w, t be vectors of W, and f be an additive wrt. second argument complex-homogeneous wrt. second argument form of V, W. Then f(⟨⟨v, w − t⟩⟩) = f(⟨⟨v, w⟩⟩) − f(⟨⟨v, t⟩⟩).

(39) Let V, W be vector spaces over ℂ_{F}, v, u be vectors of V, w, t be vectors of W, and f be a sesquilinear form of V, W. Then f(⟨⟨v−u, w−t⟩⟩) = f(⟨⟨v, w⟩⟩) − f(⟨⟨v, t⟩⟩) − (f(⟨⟨u, w⟩⟩) − f(⟨⟨u, t⟩⟩)).

(40) Let V, W be add-associative right zeroed right complementable vector space-like non empty vector space structures over ℂ_{F}, v, u be vectors of V, w, t be vectors of W, a, b be elements of the carrier of ℂ_{F}, and f be a sesquilinear form of V, W. Then f(⟨⟨v + a · u, w + b · t⟩⟩) = f(⟨⟨v, w⟩⟩) + \overline{b} · f(⟨⟨v, t⟩⟩) + (a · f(⟨⟨u, w⟩⟩) + a · (\overline{b} · f(⟨⟨u, t⟩⟩))).

(41) Let V, W be vector spaces over ℂ_{F}, v, u be vectors of V, w, t be vectors of W, a, b be elements of the carrier of ℂ_{F}, and f be a sesquilinear form of V, W. Then f(⟨⟨v − a · u, w − b · t⟩⟩) = f(⟨⟨v, w⟩⟩) − \overline{b} · f(⟨⟨v, t⟩⟩) − (a · f(⟨⟨u, w⟩⟩) − a · (\overline{b} · f(⟨⟨u, t⟩⟩))).

(42) Let V be an add-associative right zeroed right complementable vector space-like non empty vector space structure over ℂ_{F}, f be a complexhomogeneous wrt. second argument form of V, V, and v be a vector of V. Then f(⟨⟨v, 0_{V} ⟩⟩) = 0_{ℂ_{F}}.

(43) Let V be a vector space over ℂ_{F}, v, w be vectors of V, and f be a hermitan form of V. Then f(⟨⟨v, w⟩⟩)+f(⟨⟨v, w⟩⟩)+f(⟨⟨v, w⟩⟩)+f(⟨⟨v, w⟩⟩) = ((f(⟨⟨v + w, v + w⟩⟩) − f(⟨⟨v − w, v − w⟩⟩)) + i_{ℂ_{F}} · f(⟨⟨v + i_{ℂ_{F}} · w, v + i_{ℂ_{F}} · w⟩⟩)) − i_{ℂ_{F}} · f(⟨⟨v − i_{ℂ_{F}} · w, v − i_{ℂ_{F}} · w⟩⟩).

Let V be a non empty vector space structure over ℂ_{F}, let f be a form of V, V, and let v be a vector of V. The functor ||v||_{f}^{2} yields a real number and is defined as follows:

(Def. 9) ||v||_{f}^{2} = ℜ(f(⟨⟨v, v⟩⟩)).

The following propositions are true:

(44) Let V be an add-associative right zeroed right complementable vector space-like non empty vector space structure over ℂ_{F}, f be a diagonal plusreal valued diagonal real valued form of V, V, and v be a vector of V. Then |f(⟨⟨v, v⟩⟩)| = ℜ(f(⟨⟨v, v⟩⟩)) and ||v||_{f}^{2} = |f(⟨⟨v, v⟩⟩)|.

(45) Let V be a vector space over ℂ_{F}, v, w be vectors of V, f be a sesquilinear form of V, V, r be a real number, and a be an element of the carrier of ℂ_{F}. Suppose |a| = 1 and ℜ(a · f(⟨⟨w, v⟩⟩)) = |f(⟨⟨w, v⟩⟩)| and ℑ(a · f(⟨⟨w, v⟩⟩)) = 0. Then f(⟨⟨v − (r + 0i_{ℂ_{F}}) · a · w, v − (r + 0i_{ℂ_{F}}) · a · w⟩⟩) = (f(⟨⟨v, v⟩⟩)−(r+0i_{ℂ_{F}})·(a·f(⟨⟨w, v⟩⟩))−(r+0i_{ℂ_{F}})·(\overline{a}·f(⟨⟨v, w⟩⟩)))+(r^{2}+0i_{ℂ_{F}})·f(⟨⟨w, w⟩⟩).

(46) Let V be a vector space over ℂ_{F}, v, w be vectors of V, f be a diagonal plus-real valued hermitan form of V, r be a real number, and a be an element of the carrier of ℂ_{F}. Suppose |a| = 1 and ℜ(a · f(⟨⟨w, v⟩⟩)) = |f(⟨⟨w, v⟩⟩)| and ℑ(a · f(⟨⟨w, v⟩⟩)) = 0. Then ℜ(f(⟨⟨v − (r + 0i_{ℂ_{F}}) · a · w, v − (r + 0i_{ℂ_{F}}) · a · w⟩⟩)) = (||v||_{f}^{2} − 2 · |f(⟨⟨w, v⟩⟩)| · r) + ||w||_{f}^{2} · r^{2} and 0 (||v||_{f}^{2} − 2 · |f(⟨⟨w, v⟩⟩)| · r) + ||w||_{f}^{2} · r^{2}.

(47) Let V be a vector space over ℂ_{F}, v, w be vectors of V, and f be a diagonal plus-real valued hermitan form of V. If ||w||_{f}^{2} = 0, then |f(⟨⟨w, v⟩⟩)| = 0.

(48) Let V be a vector space over ℂ_{F}, v, w be vectors of V, and f be a diagonal plus-real valued hermitan form of V. Then |f(⟨⟨v, w⟩⟩)|^{2} ||v||_{f}^{2} · ||w||_{f}^{2}.

(49) Let V be a vector space over ℂ_{F}, f be a diagonal plus-real valued hermitan form of V, and v, w be vectors of V. Then |f(⟨⟨v, w⟩⟩)|^{2} |f(⟨⟨v, v⟩⟩)| · |f(⟨⟨w, w⟩⟩)|.

(50) Let V be a vector space over ℂ_{F}, f be a diagonal plus-real valued hermitan form of V, and v, w be vectors of V. Then ||v + w||_{f}^{2}

(\sqrt{||v||_{f}^{2}} + \sqrt{||w||_{f}^{2}})^{2}.

(51) Let V be a vector space over ℂ_{F}, f be a diagonal plus-real valued hermitan form of V, and v, w be vectors of V. Then |f(⟨⟨v + w, v + w⟩⟩)| (\sqrt{|f(⟨⟨v, v⟩⟩)|} + \sqrt{|f(⟨⟨w, w⟩⟩)|})^{2}.

(52) Let V be a vector space over ℂ_{F}, f be a hermitan form of V, and v, w be elements of the carrier of V. Then ||v+w||_{f}^{2} +||v−w||_{f}^{2} = 2·||v||_{f}^{2} +2·||w||_{f}^{2}.

(53) Let V be a vector space over ℂ_{F}, f be a diagonal plus-real valued hermitan form of V, and v, w be elements of the carrier of V. Then |f(⟨⟨v + w, v + w⟩⟩)| + |f(⟨⟨v − w, v − w⟩⟩)| = 2 · |f(⟨⟨v, v⟩⟩)| + 2 · |f(⟨⟨w, w⟩⟩)|.

Let V be a non empty vector space structure over ℂ_{F} and let f be a form of V, V. The functor || · ||_{f} yields a RFunctional of V and is defined as follows:

(Def. 10) For every element v of the carrier of V holds (|| · ||_{f})(v) = \sqrt{||v||_{f}^{2}}.

Let V be a vector space over ℂ_{F} and let f be a diagonal plus-real valued hermitan form of V. Then || · ||_{f} is a Semi-Norm of V.

4. Kernel of Hermitan Forms and Hermitan Forms in Quotient Vector Spaces

Let V be an add-associative right zeroed right complementable vector space-like non empty vector space structure over ℂ_{F} and let f be a complexhomogeneous wrt. second argument form of V, V. Note that diagkerf is non empty.

We now state several propositions:

(54) Let V be a vector space over ℂ_{F} and f be a diagonal plus-real valued hermitan form of V. Then diagkerf is linearly closed.

(55) For every vector space V over ℂ_{F} and for every diagonal plus-real valued hermitan form f of V holds diagkerf = leftkerf.

(56) For every vector space V over ℂ_{F} and for every diagonal plus-real valued hermitan form f of V holds diagkerf = rightkerf.

(57) For every non empty vector space structure V over ℂ_{F} and for every form f of V, V holds diagkerf = diagker \overline{f} .

(58) For all non empty vector space structures V, W over ℂ_{F} and for every form f of V, W holds leftkerf = leftker \overline{f} and rightkerf = rightker \overline{f} .

(59) For every vector space V over ℂ_{F} and for every diagonal plus-real valued hermitan form f of V holds LKerf = RKer \overline{f} .

(60) Let V be a vector space over ℂ_{F}, f be a diagonal plus-real valued diagonal real valued form of V, V, and v be a vector of V. If ℜ(f(⟨⟨v, v⟩⟩)) = 0, then f(⟨⟨v, v⟩⟩) = 0_{ℂ_{F}}.

(61) Let V be a vector space over ℂ_{F}, f be a diagonal plus-real valued hermitan form of V, and v be a vector of V. Suppose ℜ(f(⟨⟨v, v⟩⟩)) = 0 and f is non degenerated on left and non degenerated on right. Then v = 0_{V} .

Let V be a non empty vector space structure over ℂ_{F}, let W be a vector space over ℂ_{F}, and let f be an additive wrt. second argument complex-homogeneous wrt. second argument form of V, W. The functor RQForm^{∗}(f) yielding an additive wrt. second argument complex-homogeneous wrt. second argument form of V, ^{W} /_{RKer \overline{f}} is defined as follows:

(Def. 11) RQForm^{∗}(f) = \overline{RQForm(\overline{f})}.

We now state the proposition

(62) Let V be a non empty vector space structure over ℂ_{F}, W be a vector space over ℂ_{F}, f be an additive wrt. second argument complexhomogeneous wrt. second argument form of V, W, v be a vector of V, and w be a vector of W. Then (RQForm^{∗}(f))(⟨⟨v, w + RKer \overline{f} ⟩⟩) = f(⟨⟨v, w⟩⟩).

Let V, W be vector spaces over ℂ_{F} and let f be a sesquilinear form of V, W. Note that LQForm(f) is additive wrt. second argument and complexhomogeneous wrt. second argument and RQForm^{∗}(f) is additive wrt. first argument and homogeneous wrt. first argument.

Let V, W be vector spaces over ℂ_{F} and let f be a sesquilinear form of V, W. The functor QForm^{∗} f yields a sesquilinear form of ^{V} /_{LKer f}, ^{W} /_{RKer \overline{f}} and is defined by the condition (Def. 12).

(Def. 12) Let A be a vector of ^{V} /_{LKer f}, B be a vector of ^{W} /_{RKer \overline{f}}, v be a vector of V, and w be a vector of W. If A = v + LKerf and B = w + RKer \overline{f} , then (QForm^{∗} f)(⟨⟨A, B⟩⟩) = f(⟨⟨v, w⟩⟩).

Let V, W be non trivial vector spaces over ℂ_{F} and let f be a non constant sesquilinear form of V, W. Observe that QForm^{∗} f is non constant.

Let V be a right zeroed non empty vector space structure over ℂ_{F}, let W be a vector space over ℂ_{F}, and let f be an additive wrt. second argument complex-homogeneous wrt. second argument form of V, W. One can verify that RQForm^{∗}(f) is non degenerated on right.

One can prove the following propositions:

(63) Let V be a non empty vector space structure over ℂ_{F}, W be a vector space over ℂ_{F}, and f be an additive wrt. second argument complexhomogeneous wrt. second argument form of V, W. Then leftkerf = leftker(RQForm^{∗}(f)).

(64) For all vector spaces V, W over ℂ_{F} and for every sesquilinear form f of V, W holds RKer \overline{f} = RKer \overline{LQForm(f)}.

(65) For all vector spaces V, W over ℂ_{F} and for every sesquilinear form f of V, W holds LKerf = LKer(RQForm^{∗}(f)).

(66) For all vector spaces V, W over ℂ_{F} and for every sesquilinear form f of V, W holds QForm^{∗} f = RQForm^{∗}(LQForm(f)) and QForm^{∗} f = LQForm(RQForm^{∗}(f)).

(67) Let V, W be vector spaces over ℂ_{F} and f be a sesquilinear form of V, W. Then leftker(QForm^{∗} f) = leftker(RQForm^{∗}(LQForm(f))) and rightker(QForm^{∗} f) = rightker(RQForm^{∗}(LQForm(f))) and leftker(QForm^{∗} f) = leftker(LQForm(RQForm^{∗}(f))) and rightker(QForm^{∗} f) = rightker(LQForm(RQForm^{∗}(f))).

Let V, W be vector spaces over ℂ_{F} and let f be a sesquilinear form of V, W. Note that RQForm^{∗}(LQForm(f)) is non degenerated on left and non degenerated on right and LQForm(RQForm^{∗}(f)) is non degenerated on left and non degenerated on right.

Let V, W be vector spaces over ℂ_{F} and let f be a sesquilinear form of V, W. Note that QForm^{∗} f is non degenerated on left and non degenerated on right.

5. Scalar Product in Quotient Vector Space Generated by Non-Negative Hermitan Form

Let V be a non empty vector space structure over ℂ_{F} and let f be a form of V, V. We say that f is positive diagonal valued if and only if:

(Def. 13) For every vector v of V such that v ≠ 0_{V} holds 0 < ℜ(f(⟨⟨v, v⟩⟩)).

Let V be a right zeroed non empty vector space structure over ℂ_{F}. Note that every form of V, V which is positive diagonal valued and additive wrt. first argument is also diagonal plus-real valued.

Let V be a right zeroed non empty vector space structure over ℂ_{F}. One can verify that every form of V, V which is positive diagonal valued and additive wrt. second argument is also diagonal plus-real valued.

Let V be a vector space over ℂ_{F} and let f be a diagonal plus-real valued hermitan form of V. The functor ⟨·|·⟩_{f} yields a diagonal plus-real valued hermitan form of ^{V} /_{LKer f} and is defined as follows:

(Def. 14) ⟨·|·⟩_{f} = QForm^{∗} f.

Next we state three propositions:

(68) Let V be a vector space over ℂ_{F}, f be a diagonal plus-real valued hermitan form of V, A, B be vectors of ^{V} /_{LKer f}, and v, w be vectors of V. If A = v + LKerf and B = w + LKerf, then (⟨·|·⟩_{f})(⟨⟨A, B⟩⟩) = f(⟨⟨v, w⟩⟩).

(69) For every vector space V over ℂ_{F} and for every diagonal plus-real valued hermitan form f of V holds leftker(⟨·|·⟩_{f}) = leftker(QForm^{∗} f).

(70) For every vector space V over ℂ_{F} and for every diagonal plus-real valued

hermitan form f of V holds rightker(⟨·|·⟩_{f}) = rightker(QForm^{∗} f). Let V be a vector space over ℂ_{F} and let f be a diagonal plus-real valued her-

mitan form of V. Observe that ⟨·|·⟩_{f} is non degenerated on left, non degenerated on right, and positive diagonal valued.

Let V be a non trivial vector space over ℂ_{F} and let f be a diagonal plus-real valued non constant hermitan form of V. Note that ⟨·|·⟩_{f} is non constant.

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The Class of Series-Parallel Graphs. Part I

Krzysztof Retel University of Białystok

Summary. The paper introduces some preliminary notions concerning the graph theory according to [20]. We define Necklace n as a graph with vertex {1, 2, 3, …, n} and edge set {(1, 2), (2, 3), …, (n − 1, n)}. The goal of the article is to prove that Necklace n and Complement of Necklace n are isomorphic for n = 0, 1, 4.

MML Identifier: NECKLACE.

The terminology and notation used in this paper are introduced in the following papers: [23], [22], [25], [12], [1], [15], [5], [11], [2], [24], [26], [28], [18], [6], [7], [21], [13], [19], [27], [8], [9], [10], [17], [3], [4], [14], and [16].

1. Preliminaries

We adopt the following rules: n is a natural number and x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, y_{1}, y_{2}, y_{3} are sets.

Let x_{1}, x_{2}, x_{3}, x_{4}, x_{5} be sets. We say that x_{1}, x_{2}, x_{3}, x_{4}, x_{5} are mutually different if and only if:

(Def. 1) x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{1} ≠ x_{5} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4}

and x_{2} ≠ x_{5} and x_{3} ≠ x_{4} and x_{3} ≠ x_{5} and x_{4} ≠ x_{5}. Next we state several propositions:

(1) If x_{1}, x_{2}, x_{3}, x_{4}, x_{5} are mutually different, then card{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} = 5.

(2) 4 = {0, 1, 2, 3}.

(3) [: {x_{1}, x_{2}, x_{3}}, {y_{1}, y_{2}, y_{3}}:] = {⟨⟨x_{1}, y_{1}⟩⟩, ⟨⟨x_{1}, y_{2}⟩⟩, ⟨⟨x_{1}, y_{3}⟩⟩, ⟨⟨x_{2}, y_{1}⟩⟩, ⟨⟨x_{2}, y_{2}⟩⟩, ⟨⟨x_{2}, y_{3}⟩⟩, ⟨⟨x_{3}, y_{1}⟩⟩, ⟨⟨x_{3}, y_{2}⟩⟩, ⟨⟨x_{3}, y_{3}⟩⟩}.

(4) For every set x and for every natural number n such that x ∈ n holds x is a natural number.

(5) For every non empty natural number x holds 0 ∈ x. Let us observe that every set which is natural is also cardinal. Let X be a set. One can check that δ_{X} is one-to-one. Next we state the proposition

(6) For every set X holds \overline{\overline{△_{X}}} = \overline{\overline{X}} . Let R be a trivial binary relation. Observe that domR is trivial. Let us observe that every function which is trivial is also one-to-one. We now state several propositions:

(7) For all functions f, g such that domf misses domg holds rng(f+·g) = rngf ∪ rngg.

(8) For all one-to-one functions f, g such that domf misses domg and rngf misses rngg holds (f+·g)^{−1} = f^{−1} +·g^{−1} .

(9) For all sets A, a, b holds (A ⟼→ a)+·(A ⟼→ b) = A ⟼→ b.

(10) For all sets a, b holds (a⟼→ . b)^{−1} = b⟼→ . a.

(11) For all sets a, b, c, d such that a = b iff c = d holds [a ⟼→ c, b ⟼→ d]^{−1} =

[c ⟼→ a, d ⟼→ b]. The scheme Convers deals with a non empty set A, a binary relation B, two

unary functors F and G yielding sets, and a unary predicate P, and states that:

B^{⌣} = {⟨⟨F(x), G(x)⟩⟩; x ranges over elements of A : P[x]} provided the parameters meet the following condition:

• B = {⟨⟨G(x), F(x)⟩⟩; x ranges over elements of A : P[x]}. Next we state the proposition

(12) For all natural numbers i, j, n such that i < j and j ∈ n holds i ∈ n.

2. Auxiliary Concepts

Let R, S be non empty relational structures. We say that S embeds R if and only if the condition (Def. 2) is satisfied.

(Def. 2) There exists a map f from R into S such that (i) f is one-to-one, and (ii) for all elements x, y of the carrier of R holds ⟨⟨x, y⟩⟩ ∈ the internal

relation of R iff ⟨⟨f(x), f(y)⟩⟩∈ the internal relation of S. Let us note that the predicate S embeds R is reflexive.

One can prove the following proposition

(13) For all non empty relational structures R, S, T such that R embeds S

and S embeds T holds R embeds T. Let R, S be non empty relational structures. We say that R is equimorphic

to S if and only if: (Def. 3) R embeds S and S embeds R.

Let us notice that the predicate R is equimorphic to S is reflexive and symmetric.

The following proposition is true

(14) Let R, S, T be non empty relational structures. Suppose R is equimor-

phic to S and S is equimorphic to T. Then R is equimorphic to T. Let R be a non empty relational structure. We introduce R is parallel as an

antonym of R is connected.

Let R be a relational structure. We say that R is symmetric if and only if: (Def. 4) The internal relation of R is symmetric in the carrier of R.

Let R be a relational structure. We say that R is asymmetric if and only if: (Def. 5) The internal relation of R is asymmetric.

We now state the proposition

(15) Let R be a relational structure. Suppose R is asymmetric. Then the

internal relation of R misses (the internal relation of R)^{⌣}. Let R be a relational structure. We say that R is irreflexive if and only if:

(Def. 6) For every set x such that x ∈ the carrier of R holds ⟨⟨x, x⟩⟩ ∉ the internal

relation of R. Let n be a natural number. The functor n -SuccRelStr yielding a strict rela-

tional structure is defined as follows: (Def. 7) The carrier of n -SuccRelStr = n and the internal relation of

n -SuccRelStr = {⟨⟨i, i + 1⟩⟩; i ranges over natural numbers: i + 1 < n}. The following propositions are true:

(16) For every natural number n holds n -SuccRelStr is asymmetric.

(17) If n > 0, then \overline{\overline{the internal relation of n -SuccRelStr}} = n − 1. Let R be a relational structure. The functor SymRelStrR yielding a strict

relational structure is defined by the conditions (Def. 8). (Def. 8)(i) The carrier of SymRelStrR = the carrier of R, and

(ii) the internal relation of SymRelStrR = (the internal relation of R)∪(the

internal relation of R)^{⌣}. Let R be a relational structure. Note that SymRelStrR is symmetric. Let us mention that there exists a relational structure which is non empty

and symmetric.

Let R be a symmetric relational structure. One can verify that the internal relation of R is symmetric.

Let R be a relational structure. The functor ComplRelStrR yielding a strict relational structure is defined by the conditions (Def. 9).

(Def. 9)(i) The carrier of ComplRelStrR = the carrier of R, and

(ii) the internal relation of ComplRelStrR = (the internal relation of R)^{c} \ △_{the carrier of R}.

Let R be a non empty relational structure. Observe that ComplRelStrR is non empty.

Next we state the proposition

(18) Let S, R be relational structures. Suppose S and R are isomorphic. Then \overline{\overline{the internal relation of S}} = \overline{\overline{the internal relation of R}}.

3. Necklace n

Let n be a natural number. The functor Necklacen yielding a strict relational structure is defined as follows:

(Def. 10) Necklacen = SymRelStrn -SuccRelStr. Let n be a natural number. One can check that Necklacen is symmetric. We now state two propositions:

(19) The internal relation of Necklacen = {⟨⟨i, i + 1⟩⟩; i ranges over natural numbers: i+1 < n}∪{⟨⟨i+1, i⟩⟩; i ranges over natural numbers: i+1 < n}.

(20) Let x be a set. Then x ∈ the internal relation of Necklacen if and only if there exists a natural number i such that i + 1 < n but x = ⟨⟨i, i + 1⟩⟩or x = ⟨⟨i + 1, i⟩⟩.

Let n be a natural number. Observe that Necklacen is irreflexive.

Next we state the proposition

(21) For every natural number n holds the carrier of Necklacen = n. Let n be a non empty natural number. Observe that Necklacen is non empty. Let n be a natural number. Observe that the carrier of Necklacen is finite. One can prove the following propositions:

(22) For all natural numbers n, i such that i + 1 < n holds ⟨⟨i, i + 1⟩⟩ ∈ the internal relation of Necklacen.

(23) For every natural number n and for every natural number i such that i ∈ the carrier of Necklacen holds i < n.

(24) For every non empty natural number n holds Necklacen is connected.

(25) For all natural numbers i, j such that ⟨⟨i, j⟩⟩ ∈ the internal relation of Necklacen holds i = j + 1 or j = i + 1.

(26) Let i, j be natural numbers. Suppose i = j + 1 or j = i + 1 but i ∈ the carrier of Necklacen but j ∈ the carrier of Necklacen. Then ⟨⟨i, j⟩⟩ ∈ the internal relation of Necklacen.

(27) If n > 0, then \overline{\overline{\{⟨⟨i + 1, i⟩⟩; i ranges over natural numbers: i + 1 < n\}}} = n − 1.

(28) If n > 0, then \overline{\overline{the internal relation of Necklacen}} = 2 · (n − 1).

(29) Necklace1 and ComplRelStrNecklace1 are isomorphic.

(30) Necklace4 and ComplRelStrNecklace4 are isomorphic.

the class of series-parallel graphs. part … 103

(31) If Necklacen and ComplRelStrNecklacen are isomorphic, then n = 0 or n = 1 or n = 4.

References

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[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[4] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.

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[6] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.

[7] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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[9] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[10] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[13] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

[14] Czesław Byliński. Galois connections. Formalized Mathematics, 6(1):131–143, 1997.

[15] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[16] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501–505, 1996.

[17] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

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[19] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

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[23] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

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[25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[28] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

Term Orders

Christoph Schwarzweller University of Tuebingen

Summary. We continue the formalization of [5] towards Gröbner Bases. Here we deal with term orders, that is with orders on bags. We introduce the notions of head term, head coefficient, and head monomial necessary to define reduction for polynomials.

MML Identifier: TERMORD.

The papers [16], [21], [22], [1], [10], [23], [7], [8], [3], [2], [12], [20], [17], [4], [6], [9], [11], [24], [14], [13], [18], [19], and [15] provide the terminology and notation for this paper.

1. Preliminaries

One can check that there exists a loop structure which is non trivial.

Let us mention that there exists a non trivial loop structure which is addassociative, right complementable, and right zeroed.

Let X be a set and let b be a bag of X. We say that b is non-zero if and only if:

(Def. 1) b ≠ EmptyBagX. Next we state two propositions:

(1) For every set X and for all bags b_{1}, b_{2} of X holds b_{1} | b_{2} iff there exists a bag b of X such that b_{2} = b_{1} + b.

(2) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and p be a series of n, L. Then 0 (n, L) ∗ p = 0 (n, L).

Let n be an ordinal number, let L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and let m_{1}, m_{2} be monomials of n, L. Note that m_{1} ∗ m_{2} is monomial-like.

Let n be an ordinal number, let L be an add-associative right complementable right zeroed distributive non empty double loop structure, and let c_{1}, c_{2} be constant polynomials of n, L. One can verify that c_{1} ∗ c_{2} is constant.

One can prove the following two propositions:

(3) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, b, b′ be bags of n, and a, a′ be non-zero elements of L. Then Monom(a · a′, b + b′) = Monom(a, b) ∗ Monom(a′, b′).

(4) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, and a, a′ be elements of L. Then a · a′ (n, L) = (a (n, L)) ∗ (a′ (n, L)).

2. Term Orders

Let n be an ordinal number. One can verify that there exists a term order of n which is admissible and connected.

Let n be a natural number. Observe that every admissible term order of n is well founded.

Let n be an ordinal number, let T be a term order of n, and let b, b′be bags of n. The predicate b _{T} b′ is defined by:

(Def. 2) ⟨⟨b, b′⟩⟩∈ T.

Let n be an ordinal number, let T be a term order of n, and let b, b′be bags of n. The predicate b <_{T} b′ is defined by:

(Def. 3) b _{T} b′ and b ≠ b′.

Let n be an ordinal number, let T be a term order of n, and let b_{1}, b_{2} be bags of n. The functor min_{T} (b_{1}, b_{2}) yields a bag of n and is defined as follows:

(Def. 4) min_{T} (b_{1}, b_{2}) = \{ \begin{matrix}b_{1}, if b_{1} _{T} b_{2}, \\ b_{2}, otherwise. \end{matrix}

The functor max_{T} (b_{1}, b_{2}) yields a bag of n and is defined as follows:

(Def. 5) max_{T} (b_{1}, b_{2}) = \{ \begin{matrix}b_{1}, if b_{2} _{T} b_{1}, \\ b_{2}, otherwise. \end{matrix}

We now state a number of propositions:

(5) Let n be an ordinal number, T be a connected term order of n, and b_{1}, b_{2} be bags of n. Then b_{1} _{T} b_{2} if and only if b_{2} ≮_{T} b_{1}.

(6) For every ordinal number n and for every term order T of n and for every bag b of n holds b _{T} b.

(7) Let n be an ordinal number, T be a term order of n, and b_{1}, b_{2} be bags of n. If b_{1} _{T} b_{2} and b_{2} _{T} b_{1}, then b_{1} = b_{2}.

(8) Let n be an ordinal number, T be a term order of n, and b_{1}, b_{2}, b_{3} be bags of n. If b_{1} _{T} b_{2} and b_{2} _{T} b_{3}, then b_{1} _{T} b_{3}.

(9) For every ordinal number n and for every admissible term order T of n and for every bag b of n holds EmptyBagn _{T} b.

(10) Let n be an ordinal number, T be an admissible term order of n, and b_{1}, b_{2} be bags of n. If b_{1} | b_{2}, then b_{1} _{T} b_{2}.

(11) For every ordinal number n and for every term order T of n and for all bags b_{1}, b_{2} of n holds min_{T} (b_{1}, b_{2}) = b_{1} or min_{T} (b_{1}, b_{2}) = b_{2}.

(12) For every ordinal number n and for every term order T of n and for all bags b_{1}, b_{2} of n holds max_{T} (b_{1}, b_{2}) = b_{1} or max_{T} (b_{1}, b_{2}) = b_{2}.

(13) Let n be an ordinal number, T be a connected term order of n, and b_{1}, b_{2} be bags of n. Then min_{T} (b_{1}, b_{2}) _{T} b_{1} and min_{T} (b_{1}, b_{2}) _{T} b_{2}.

(14) Let n be an ordinal number, T be a connected term order of n, and b_{1}, b_{2} be bags of n. Then b_{1} _{T} max_{T} (b_{1}, b_{2}) and b_{2} _{T} max_{T} (b_{1}, b_{2}).

(15) Let n be an ordinal number, T be a connected term order of n, and b_{1}, b_{2} be bags of n. Then min_{T} (b_{1}, b_{2}) = min_{T} (b_{2}, b_{1}) and max_{T} (b_{1}, b_{2}) = max_{T} (b_{2}, b_{1}).

(16) Let n be an ordinal number, T be a connected term order of n, and b_{1}, b_{2} be bags of n. Then min_{T} (b_{1}, b_{2}) = b_{1} if and only if max_{T} (b_{1}, b_{2}) = b_{2}.

3. Head Terms, Head Monomials, and Head Coefficients

Let n be an ordinal number, let T be a connected term order of n, let L be a non empty zero structure, and let p be a polynomial of n, L. The functor HT(p, T) yields an element of Bagsn and is defined as follows:

(Def. 6) Supportp = ∅ and HT(p, T) = EmptyBagn or HT(p, T) ∈ Supportp

and for every bag b of n such that b ∈ Supportp holds b _{T} HT(p, T). Let n be an ordinal number, let T be a connected term order of n, let L

be a non empty zero structure, and let p be a polynomial of n, L. The functor HC(p, T) yielding an element of L is defined as follows:

(Def. 7) HC(p, T) = p(HT(p, T)).

Let n be an ordinal number, let T be a connected term order of n, let L be a non empty zero structure, and let p be a polynomial of n, L. The functor HM(p, T) yielding a monomial of n, L is defined by:

(Def. 8) HM(p, T) = Monom(HC(p, T), HT(p, T)).

Let n be an ordinal number, let T be a connected term order of n, let L be a non trivial zero structure, and let p be a non-zero polynomial of n, L. Observe that HM(p, T) is non-zero and HC(p, T) is non-zero.

The following propositions are true:

(17) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, and p be a polynomial of n, L. Then HC(p, T) = 0_{L} if and only if p = 0 (n, L).

(18) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, and p be a polynomial of n, L. Then (HM(p, T))(HT(p, T)) = p(HT(p, T)).

(19) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, p be a polynomial of n, L, and b be a bag of n. If b ≠ HT(p, T), then (HM(p, T))(b) = 0_{L}.

(20) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, and p be a polynomial of n, L. Then SupportHM(p, T) ⊆ Supportp.

(21) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, and p be a polynomial of n, L. Then SupportHM(p, T) = ∅ or SupportHM(p, T) = {HT(p, T)}.

(22) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, and p be a polynomial of n, L. Then termHM(p, T) = HT(p, T) and coefficientHM(p, T) = HC(p, T).

(23) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, and m be a monomial of n, L. Then HT(m, T) = termm and HC(m, T) = coefficientm and HM(m, T) = m.

(24) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, and c be a constant polynomial of n, L. Then HT(c, T) = EmptyBagn and HC(c, T) = c(EmptyBagn).

(25) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, and a be an element of L. Then HT(a (n, L), T) = EmptyBagn and HC(a (n, L), T) = a.

(26) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, and p be a polynomial of n, L. Then HT(HM(p, T), T) = HT(p, T).

(27) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, and p be a polynomial of n, L. Then HC(HM(p, T), T) = HC(p, T).

(28) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, and p be a polynomial of n, L. Then HM(HM(p, T), T) = HM(p, T).

(29) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable left zeroed right zeroed unital distributive integral domain-like non trivial double loop structure, and p, q be non-zero polynomials of n, L. Then HT(p, T) + HT(q, T) ∈

Support(p ∗ q).

(30) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and p, q be polynomials of n, L. Then Support(p ∗ q) ⊆ {s + t; s ranges over elements of Bagsn, t ranges over elements of Bagsn : s ∈ Supportp ∧ t ∈ Supportq}.

(31) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, and p, q be non-zero polynomials of n, L. Then HT(p∗q, T) = HT(p, T)+HT(q, T).

(32) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, and p, q be non-zero polynomials of n, L. Then HC(p∗q, T) = HC(p, T)·HC(q, T).

(33) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, and p, q be non-zero polynomials of n, L. Then HM(p∗q, T) = HM(p, T)∗HM(q, T).

(34) Let n be an ordinal number, T be an admissible connected term order of n, L be a right zeroed non empty loop structure, and p, q be polynomials of n, L. Then HT(p + q, T) _{T} max_{T} (HT(p, T), HT(q, T)).

4. Reductum of a Polynomial

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed non empty loop structure, and let p be a polynomial of n, L. The functor Red(p, T) yielding a polynomial of n, L is defined by:

(Def. 9) Red(p, T) = p − HM(p, T).

The following propositions are true:

(35) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p be a polynomial of n, L. Then SupportRed(p, T) ⊆ Supportp.

(36) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p be a polynomial of n, L. Then SupportRed(p, T) = Supportp \ {HT(p, T)}.

(37) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop 110 christoph schwarzweller

structure, and p be a polynomial of n, L. Then Support(HM(p, T) + Red(p, T)) = Supportp.

(38) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p be a polynomial of n, L. Then HM(p, T) + Red(p, T) = p.

(39) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p be a polynomial of n, L. Then (Red(p, T))(HT(p, T)) = 0_{L}.

(40) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, p be a polynomial of n, L, and b be a bag of n. If b ∈ Supportp and b ≠ HT(p, T), then (Red(p, T))(b) = p(b).

Acknowledgments

I’d like to thank Piotr Rudnicki for valuable discussions on how to represent term orders in Mizar.

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Polynomial Reduction

Christoph Schwarzweller University of Tuebingen

Summary. We continue the formalization of [8] towards Gröbner Bases. In this article we introduce reduction of polynomials and prove its termination, its adequateness for ideal congruence as well as the translation lemma used later to show confluence of reduction.

MML Identifier: POLYRED.

The notation and terminology used here are introduced in the following papers: [21], [26], [12], [27], [29], [28], [10], [11], [4], [3], [17], [6], [22], [13], [5], [25], [2], [7], [24], [9], [16], [14], [19], [1], [23], [18], [15], and [20].

1. Preliminaries

Let n be an ordinal number and let R be a non trivial zero structure. One can verify that there exists a monomial of n, R which is non-zero.

Let us observe that there exists a field which is non trivial.

Let us note that every left zeroed add-right-cancelable right distributive left unital commutative associative non empty double loop structure which is fieldlike is also integral domain-like.

Let n be an ordinal number, let L be an add-associative right complementable left zeroed right zeroed unital distributive integral domain-like non trivial double loop structure, and let p, q be non-zero finite-Support series of n, L. Note that p ∗ q is non-zero.

2. More on Polynomials and Monomials

The following propositions are true:

(1) Let X be a set, L be an Abelian add-associative right zeroed right complementable non empty loop structure, and p, q be series of X, L. Then − (p + q) = − p + − q.

(2) For every set X and for every left zeroed non empty loop structure L and for every series p of X, L holds 0 (X, L) + p = p.

(3) Let X be a set, L be an add-associative right zeroed right complementable non empty loop structure, and p be a series of X, L. Then − p + p = 0 (X, L) and p + − p = 0 (X, L).

(4) Let n be a set, L be an add-associative right zeroed right complementable non empty loop structure, and p be a series of n, L. Then p−0 (n, L) = p.

(5) Let n be an ordinal number, L be an add-associative right complementable right zeroed add-left-cancelable left distributive non empty double loop structure, and p be a series of n, L. Then 0 (n, L) ∗ p = 0 (n, L).

(6) Let n be an ordinal number, L be an Abelian right zeroed add-associative right complementable unital distributive associative commutative non trivial double loop structure, and p, q be polynomials of n, L. Then − p ∗ q = (− p) ∗ q and − p ∗ q = p ∗ − q.

(7) Let n be an ordinal number, L be an add-associative right complementable right zeroed distributive non empty double loop structure, p be a polynomial of n, L, m be a monomial of n, L, and b be a bag of n. Then (m ∗ p)(termm + b) = m(termm) · p(b).

(8) Let X be a set, L be a right zeroed add-left-cancelable left distributive non empty double loop structure, and p be a series of X, L. Then 0_{L} · p = 0 (X, L).

(9) Let X be a set, L be an add-associative right zeroed right complementable distributive non empty double loop structure, p be a series of X, L, and a be an element of L. Then − a · p= (− a) · p and − a · p= a · − p.

(10) Let X be a set, L be a left distributive non empty double loop structure, p be a series of X, L, and a, a′be elements of L. Then a·p+a′·p = (a+a′)·p.

(11) Let X be a set, L be an associative non empty multiplicative loop with zero structure, p be a series of X, L, and a, a′ be elements of L. Then (a · a′) · p = a · (a′· p).

(12) Let n be an ordinal number, L be an add-associative right zeroed right complementable unital associative commutative distributive non empty double loop structure, p, p′ be series of n, L, and a be an element of L. Then a · (p ∗ p′) = p ∗ (a · p′).

3. Multiplication of Polynomials with Bags

Let n be an ordinal number, let b be a bag of n, let L be a non empty zero structure, and let p be a series of n, L. The functor b ∗ p yielding a series of n, L is defined as follows:

(Def. 1) For every bag b′ of n such that b | b′ holds (b ∗ p)(b′) = p(b′− ′b) and for

every bag b′ of n such that b ∤ b′ holds (b ∗ p)(b′) = 0_{L}. Let n be an ordinal number, let b be a bag of n, let L be a non empty

zero structure, and let p be a finite-Support series of n, L. Note that b ∗ p is finite-Support.

We now state a number of propositions:

(13) Let n be an ordinal number, b, b′ be bags of n, L be a non empty zero structure, and p be a series of n, L. Then (b ∗ p)(b′+ b) = p(b′).

(14) Let n be an ordinal number, L be a non empty zero structure, p be a polynomial of n, L, and b be a bag of n. Then Support(b ∗ p) ⊆ {b + b′; b′ ranges over elements of Bagsn : b′∈ Supportp}.

(15) Let n be an ordinal number, T be a connected admissible term order of n, L be a non trivial zero structure, p be a non-zero polynomial of n, L, and b be a bag of n. Then HT(b ∗ p, T) = b + HT(p, T).

(16) Let n be an ordinal number, T be a connected admissible term order of n, L be a non empty zero structure, p be a polynomial of n, L, and b, b′ be bags of n. If b′∈ Support(b ∗ p), then b′ _{T} b + HT(p, T).

(17) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, and p be a series of n, L. Then EmptyBagn ∗ p = p.

(18) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, p be a series of n, L, and b_{1}, b_{2} be bags of n. Then (b_{1} + b_{2}) ∗ p = b_{1} ∗ (b_{2} ∗ p).

(19) Let n be an ordinal number, L be an add-associative right zeroed right complementable distributive non trivial double loop structure, p be a polynomial of n, L, and a be an element of L. Then Support(a·p) ⊆ Supportp.

(20) Let n be an ordinal number, L be an integral domain-like non trivial double loop structure, p be a polynomial of n, L, and a be a non-zero element of L. Then Supportp ⊆ Support(a · p).

(21) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable distributive integral domain-like non trivial double loop structure, p be a polynomial of n, L, and a be a non-zero element of L. Then HT(a · p, T) = HT(p, T).

(22) Let n be an ordinal number, L be an add-associative right complementable right zeroed distributive non trivial double loop structure, p be a series of n, L, b be a bag of n, and a be an element of L. Then a · (b ∗ p) = Monom(a, b) ∗ p.

(23) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, p be a non-zero polynomial of n, L, q be a polynomial of n, L, and m be a non-zero monomial of n, L. If HT(p, T) ∈ Supportq, then HT(m ∗ p, T) ∈ Support(m ∗ q).

4. Orders on Polynomials

Let n be an ordinal number and let T be a connected term order of n. Observe that ⟨Bagsn, T ⟩ is connected.

Let n be a natural number and let T be an admissible term order of n. Note that ⟨Bagsn, T ⟩ is well founded.

Let n be an ordinal number, let T be a connected term order of n, let L be a non empty zero structure, and let p, q be polynomials of n, L. The predicate p _{T} q is defined as follows:

(Def. 2) ⟨⟨Supportp, Supportq⟩⟩∈ FinOrd⟨Bagsn, T ⟩.

Let n be an ordinal number, let T be a connected term order of n, let L be a non empty zero structure, and let p, q be polynomials of n, L. The predicate p <_{T} q is defined as follows:

(Def. 3) p _{T} q and Supportp ≠ Supportq.

Let n be an ordinal number, let T be a connected term order of n, let L be a non empty zero structure, and let p be a polynomial of n, L. The functor Support(p, T) yielding an element of Fin(the carrier of ⟨Bagsn, T ⟩) is defined by:

(Def. 4) Support(p, T) = Supportp.

Next we state a number of propositions:

(24) Let n be an ordinal number, T be a connected term order of n, L be a non trivial zero structure, and p be a non-zero polynomial of n, L. Then PosetMaxSupport(p, T) = HT(p, T).

(25) Let n be an ordinal number, T be a connected term order of n, L be a non empty loop structure, and p be a polynomial of n, L. Then p _{T} p.

(26) Let n be an ordinal number, T be a connected term order of n, L be a non empty loop structure, and p, q be polynomials of n, L. Then p _{T} q and q _{T} p if and only if Supportp = Supportq.

(27) Let n be an ordinal number, T be a connected term order of n, L be a non empty loop structure, and p, q, r be polynomials of n, L. If p _{T} q and q _{T} r, then p _{T} r.

(28) Let n be an ordinal number, T be a connected term order of n, L be a non empty loop structure, and p, q be polynomials of n, L. Then p _{T} q or q _{T} p.

(29) Let n be an ordinal number, T be a connected term order of n, L be a non empty loop structure, and p, q be polynomials of n, L. Then p _{T} q if and only if q ≮_{T} p.

(30) Let n be an ordinal number, T be a connected term order of n, L be a non empty zero structure, and p be a polynomial of n, L. Then 0 (n, L) _{T} p.

(31) Let n be a natural number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed unital distributive non trivial double loop structure, and P be a non empty subset of Polynom-Ring(n, L). Then there exists a polynomial p of n, L such that p ∈ P and for every polynomial q of n, L such that q ∈ P holds p _{T} q.

(32) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p, q be polynomials of n, L. Then p <_{T} q if and only if one of the following conditions is satisfied:

(i) p = 0 (n, L) and q ≠ 0 (n, L), or (ii) HT(p, T) <_{T} HT(q, T), or (iii) HT(p, T) = HT(q, T) and Red(p, T) <_{T} Red(q, T).

(33) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p be a non-zero polynomial of n, L. Then Red(p, T) <_{T} HM(p, T).

(34) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p be a polynomial of n, L. Then HM(p, T) _{T} p.

(35) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, and p be a non-zero polynomial of n, L. Then Red(p, T) <_{T} p.

5. Polynomial Reduction

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, let f, p, g be polynomials of n, L, and let b be a bag of n. We say that f reduces to g, p, b, T if and only if:

(Def. 5) f ≠ 0 (n, L) and p ≠ 0 (n, L) and b ∈ Supportf and there exists a bag s of n such that s + HT(p, T) = b and g = f − \frac{f(b)}{HC(p,T)} · (s ∗ p).

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let f, p, g be polynomials of n, L. We say that f reduces to g, p, T if and only if:

(Def. 6) There exists a bag b of n such that f reduces to g, p, b, T.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, let f, g be polynomials of n, L, and let P be a subset of Polynom-Ring(n, L). We say that f reduces to g, P, T if and only if:

(Def. 7) There exists a polynomial p of n, L such that p ∈ P and f reduces to g,

p, T. Let n be an ordinal number, let T be a connected term order of n, let L be

an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let f, p be polynomials of n, L. We say that f is reducible wrt p, T if and only if:

(Def. 8) There exists a polynomial g of n, L such that f reduces to g, p, T. We introduce f is irreducible wrt p, T and f is in normal form wrt p, T as antonyms of f is reducible wrt p, T.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, let f be a polynomial of n, L, and let P be a subset of Polynom-Ring(n, L). We say that f is reducible wrt P, T if and only if:

(Def. 9) There exists a polynomial g of n, L such that f reduces to g, P, T. We introduce f is irreducible wrt P, T and f is in normal form wrt P, T as antonyms of f is reducible wrt P, T.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let f, p, g be polynomials of n, L. We say that f top reduces to g, p, T if and only if:

(Def. 10) f reduces to g, p, HT(f, T), T.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let f, p be polynomials of n, L. We say that f is top reducible wrt p, T if and only if:

(Def. 11) There exists a polynomial g of n, L such that f top reduces to g, p, T.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, let f be a polynomial of n, L, and let P be a subset of Polynom-Ring(n, L). We say that f is top reducible wrt P, T if and only if:

(Def. 12) There exists a polynomial p of n, L such that p ∈ P and f is top reducible

wrt p, T. Next we state several propositions:

(36) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, f be a polynomial of n, L, and p be a non-zero polynomial of n, L. Then f is reducible wrt p, T if and only if there exists a bag b of n such that b ∈ Supportf and HT(p, T) | b.

(37) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and p be a polynomial of n, L. Then 0 (n, L) is irreducible wrt p, T.

(38) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, f, p be polynomials of n, L, and m be a nonzero monomial of n, L. If f reduces to f −m∗p, p, T, then HT(m∗p, T) ∈ Supportf.

(39) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, f, p, g be polynomials of n, L, and b be a bag of n. If f reduces to g, p, b, T, then b ∉ Supportg.

(40) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, f, p, g be polynomials of n, L, and b, b′ be bags of n. Suppose b <_{T} b′. If f reduces to g, p, b, T, then b′∈ Supportg iff b′∈ Supportf.

(41) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, f, p, g be polynomials of n, L, and b, b′be bags of n. If b <_{T} b′, then if f reduces to g, p, b, T, then f(b′) = g(b′).

(42) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and f, p, g be polynomials of n, L. Suppose f reduces to g, p, T. Let b be a bag of n. If b ∈ Supportg, then b _{T} HT(f, T).

(43) Let n be an ordinal number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and f, p, g be polynomials of n, L. If f reduces to g, p, T, then g <_{T} f.

6. Polynomial Reduction Relation

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let P be a subset of Polynom-Ring(n, L). The functor PolyRedRel(P, T) yields a relation between (the carrier of Polynom-Ring(n, L)) \ {0 (n, L)} and the carrier of Polynom-Ring(n, L) and is defined by:

(Def. 13) For all polynomials p, q of n, L holds ⟨⟨p, q⟩⟩ ∈ PolyRedRel(P, T) iff p

reduces to q, P, T. Next we state the proposition

(44) Let n be an ordinal number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, f, g be polynomials of n, L, and P be a subset of Polynom-Ring(n, L). If PolyRedRel(P, T) reduces f to g, then g _{T} f but g = 0 (n, L) or HT(g, T) _{T} HT(f, T).

Let n be a natural number, let T be a connected admissible term order of n, let L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and let P be a subset of Polynom-Ring(n, L). One can verify that PolyRedRel(P, T) is strongly-normalizing.

One can prove the following propositions:

(45) Let n be a natural number, T be an admissible connected term order of n, L be an add-associative right complementable left zeroed right zeroed commutative associative well unital distributive Abelian field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f, h be polynomials of n, L. If f ∈ P, then PolyRedRel(P, T) reduces h ∗ f to 0 (n, L).

(46) Let n be an ordinal number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, P be a subset of Polynom-Ring(n, L), f, g be polynomials of n, L, and m be a non-zero monomial of n, L. If f reduces to g, P, T, then m ∗ f reduces to m ∗ g, P, T.

(47) Let n be an ordinal number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, P be a subset of Polynom-Ring(n, L), f, g be polynomials of n, L, and m be a monomial of n, L. If PolyRedRel(P, T) reduces f to g, then PolyRedRel(P, T) reduces m ∗ f to m ∗ g.

(48) Let n be an ordinal number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, P be a subset of Polynom-Ring(n, L), f be a polynomial of n, L, and m be a monomial of n, L. If PolyRedRel(P, T) reduces f to 0 (n, L), then PolyRedRel(P, T) reduces m ∗ f to 0 (n, L).

(49) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f, g, h, h_{1} be polynomials of n, L. Suppose f − g = h and PolyRedRel(P, T) reduces h to h_{1}. Then there exist polynomials f_{1}, g_{1} of n, L such that f_{1} − g_{1} = h_{1} and PolyRedRel(P, T) reduces f to f_{1} and PolyRedRel(P, T) reduces g to g_{1}.

(50) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f, g be polynomials of n, L. Suppose PolyRedRel(P, T) reduces f − g to 0 (n, L). Then f and g are convergent w.r.t. PolyRedRel(P, T).

(51) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f, g be polynomials of n, L. Suppose PolyRedRel(P, T) reduces f − g to 0 (n, L). Then f and g are convertible w.r.t. PolyRedRel(P, T).

Let R be a non empty loop structure, let I be a subset of R, and let a, b be elements of R. The predicate a ≡ b(modI) is defined as follows:

(Def. 14) a − b ∈ I.

One can prove the following propositions:

(52) Let R be an add-associative left zeroed right zeroed right complementable right distributive non empty double loop structure, I be a right ideal

non empty subset of R, and a be an element of R. Then a ≡ a(modI).

(53) Let R be an add-associative right zeroed right complementable right unital right distributive non empty double loop structure, I be a right ideal non empty subset of R, and a, b be elements of R. If a ≡ b(modI), then b ≡ a(modI).

(54) Let R be an add-associative right zeroed right complementable non empty loop structure, I be an add closed non empty subset of R, and a, b, c be elements of R. If a ≡ b(modI) and b ≡ c(modI), then a ≡ c(modI).

(55) Let R be an Abelian add-associative right zeroed right complementable unital distributive associative non trivial double loop structure, I be an add closed non empty subset of R, and a, b, c, d be elements of R. If a ≡ b(modI) and c ≡ d(modI), then a + c ≡ b + d(modI).

(56) Let R be an add-associative right zeroed right complementable commutative distributive non empty double loop structure, I be an add closed right ideal non empty subset of R, and a, b, c, d be elements of R. If a ≡ b(modI) and c ≡ d(modI), then a · c ≡ b · d(modI).

(57) Let n be an ordinal number, T be a connected term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f, g be elements of Polynom-Ring(n, L). If f and g are convertible w.r.t. PolyRedRel(P, T), then f ≡ g(modP –ideal).

(58) Let n be a natural number, T be an admissible connected term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, P be a non empty subset of Polynom-Ring(n, L), and f, g be elements of Polynom-Ring(n, L). If f ≡ g(modP –ideal), then f and g are convertible w.r.t. PolyRedRel(P, T).

(59) Let n be an ordinal number, T be a connected term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f, g be polynomials of n, L. If PolyRedRel(P, T) reduces f to g, then f − g ∈ P –ideal.

(60) Let n be an ordinal number, T be a connected term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f be a polynomial of n, L. If PolyRedRel(P, T) reduces f to 0 (n, L), then f ∈ P –ideal. polynomial reduction 123

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[13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[14] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[15] Gilbert Lee and Piotr Rudnicki. On ordering of bags. Formalized Mathematics, 10(1):39– 46, 2002.

[16] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[17] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

[18] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[19] Christoph Schwarzweller. More on multivariate polynomials: Monomials and constant polynomials. Formalized Mathematics, 9(4):849–855, 2001.

[20] Christoph Schwarzweller. Term orders. Formalized Mathematics, 11(1):105–111, 2003.

[21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[22] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

[23] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

[24] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[25] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[28] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[29] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

Processes in Petri Nets

Grzegorz Bancerek Białystok Technical University

Mitsuru Aoki Shinshu University Nagano

Akio Matsumoto Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. Sequential and concurrent compositions of processes in Petri nets are introduced. A process is formalized as a set of (possible), so called, firing sequences. In the definition of the sequential composition the standard concatenation is used

R_{1} before R_{2} = {p_{1} ^{⌢} p_{2} : p_{1} ∈ R_{1} ∧ p_{2} ∈ R_{2}}

The definition of the concurrent composition is more complicated

R_{1} concur R_{2} = {q_{1} ∪ q_{2} : q_{1} misses q_{2} ∧ Seq q_{1} ∈ R_{1} ∧ Seq q_{2} ∈ R_{2}}

For example,

{⟨t_{0}⟩} concur{⟨t_{1}, t_{2}⟩} = {⟨t_{0}, t_{1}, t_{2}⟩, ⟨t_{1}, t_{0}, t_{2}⟩, ⟨t_{1}, t_{2}, t_{0}⟩}

The basic properties of the compositions are shown.

MML Identifier: PNPROC 1.

The notation and terminology used in this paper are introduced in the following papers: [14], [13], [18], [6], [17], [9], [1], [3], [7], [12], [2], [10], [15], [5], [16], [8], [11], and [4].

1. Preliminaries

We adopt the following rules: i is a natural number and x, x_{1}, x_{2}, y_{1}, y_{2} are sets.

Next we state three propositions:

(1) If i > 0, then {⟨⟨i, x⟩⟩} is a finite subsequence.

(2) For every finite subsequence q holds q = ∅ iff Seqq = ∅.

(3) For every finite subsequence q such that q = {⟨⟨i, x⟩⟩} holds Seqq = ⟨x⟩. Let us observe that every finite subsequence is finite. We now state several propositions:

(4) For every finite subsequence q such that Seqq = ⟨x⟩ there exists i such that q = {⟨⟨i, x⟩⟩}.

(5) If {⟨⟨x_{1}, y_{1}⟩⟩, ⟨⟨x_{2}, y_{2}⟩⟩} is a finite sequence, then x_{1} = 1 and x_{2} = 1 and y_{1} = y_{2} or x_{1} = 1 and x_{2} = 2 or x_{1} = 2 and x_{2} = 1.

(6) ⟨x_{1}, x_{2}⟩ = {⟨⟨1, x_{1}⟩⟩, ⟨⟨2, x_{2}⟩⟩}.

(7) For every finite subsequence p holds \overline{\overline{p}} = lenSeqp.

(8) For all binary relations P, R such that domP misses domR holds P misses R.

(9) For all sets X, Y and for all binary relations P, R such that X misses Y holds P ↾X misses R↾Y.

(10) For all functions f, g, h such that f ⊆ h and g ⊆ h and f misses g holds domf misses domg.

(11) For every set Y and for every binary relation R holds Y ↾R ⊆ R↾R^{−1} (Y).

(12) For every set Y and for every function f holds Y ↾f = f↾f^{−1} (Y).

2. Markings on Petri Nets

Let P be a set. A function is called a marking of P if: (Def. 1) domit = P and rngit ⊆ ℕ.

We adopt the following convention: P, p, x denote sets, m_{1}, m_{2}, m_{3}, m_{4}, m denote markings of P, and i, j, j_{1}, k denote natural numbers.

Let P be a set, let m be a marking of P, and let p be a set. Then m(p) is a natural number. We introduce the m multitude of p as a synonym of m(p).

The scheme MarkingLambda deals with a set A and a unary functor F yielding a natural number, and states that:

There exists a marking m of A such that for every p such that p ∈ A holds the m multitude of p = F(p)

for all values of the parameters.

Let us consider P, m_{1}, m_{2}. Let us observe that m_{1} = m_{2} if and only if:

(Def. 2) For every p such that p ∈ P holds the m_{1} multitude of p = the m_{2}

multitude of p. Let us consider P. The functor {}_{P} yielding a marking of P is defined by:

(Def. 3) {}_{P} = P ⟼→ 0.

Let P be a set and let m_{1}, m_{2} be markings of P. The predicate m_{1} ⊆ m_{2} is defined by:

(Def. 4) For every set p such that p ∈ P holds the m_{1} multitude of p the m_{2}

multitude of p. Let us note that the predicate m_{1} ⊆ m_{2} is reflexive.

We now state two propositions:

(13) {}_{P} ⊆ m.

(14) If m_{1} ⊆ m_{2} and m_{2} ⊆ m_{3}, then m_{1} ⊆ m_{3}.

Let P be a set and let m_{1}, m_{2} be markings of P. The functor m_{1} +m_{2} yields a marking of P and is defined as follows:

(Def. 5) For every set p such that p ∈ P holds the m_{1} + m_{2} multitude of p = (the

m_{1} multitude of p) + (the m_{2} multitude of p). Let us notice that the functor m_{1} + m_{2} is commutative.

The following proposition is true

(15) m + {}_{P} = m.

Let P be a set and let m_{1}, m_{2} be markings of P. Let us assume that m_{2} ⊆ m_{1}. The functor m_{1} − m_{2} yielding a marking of P is defined by:

(Def. 6) For every set p such that p ∈ P holds the m_{1} − m_{2} multitude of p = (the

m_{1} multitude of p) − (the m_{2} multitude of p). One can prove the following propositions:

(16) m_{1} ⊆ m_{1} + m_{2}.

(17) m − {}_{P} = m.

(18) If m_{1} ⊆ m_{2} and m_{2} ⊆ m_{3}, then m_{3} − m_{2} ⊆ m_{3} − m_{1}.

(19) (m_{1} + m_{2}) − m_{2} = m_{1}.

(20) If m ⊆ m_{1} and m_{1} ⊆ m_{2}, then m_{1} − m ⊆ m_{2} − m.

(21) If m_{1} ⊆ m_{2}, then (m_{2} + m_{3}) − m_{1} = (m_{2} − m_{1}) + m_{3}.

(22) If m_{1} ⊆ m_{2} and m_{2} ⊆ m_{1}, then m_{1} = m_{2}.

(23) (m_{1} + m_{2}) + m_{3} = m_{1} + (m_{2} + m_{3}).

(24) If m_{1} ⊆ m_{2} and m_{3} ⊆ m_{4}, then m_{1} + m_{3} ⊆ m_{2} + m_{4}.

(25) If m_{1} ⊆ m_{2}, then m_{2} − m_{1} ⊆ m_{2}.

(26) If m_{1} ⊆ m_{2} and m_{3} ⊆ m_{4} and m_{4} ⊆ m_{1}, then m_{1} − m_{4} ⊆ m_{2} − m_{3}.

(27) If m_{1} ⊆ m_{2}, then m_{2} = (m_{2} − m_{1}) + m_{1}.

(28) (m_{1} + m_{2}) − m_{1} = m_{2}.

(29) If m_{2} + m_{3} ⊆ m_{1}, then m_{1} − m_{2} − m_{3} = m_{1} − (m_{2} + m_{3}).

(30) If m_{3} ⊆ m_{2} and m_{2} ⊆ m_{1}, then m_{1} − (m_{2} − m_{3}) = (m_{1} − m_{2}) + m_{3}.

(31) m ∈ ℕ^{P} .

(32) If x ∈ ℕ^{P} , then x is a marking of P.

3. Transitions and Firing

Let us consider P. Transition of P is defined by: (Def. 7) There exist m_{1}, m_{2} such that it = ⟨⟨m_{1}, m_{2}⟩⟩.

In the sequel t, t_{1}, t_{2} denote transitions of P. Let us consider P, t. Then t_{1} is a marking of P. We introduce Pre t as a

synonym of t_{1}. t_{2} is a marking of P. We introduce Post t as a synonym of t_{2}.

Let us consider P, m, t. The functor fire(t, m) yielding a marking of P is defined by:

(Def. 8) fire(t, m) = \{ \begin{matrix}(m − Pre t) + Post t, if Pre t ⊆ m, \\ m, otherwise. \end{matrix}

The following proposition is true

(33) If Pre t_{1} + Pre t_{2} ⊆ m, then fire(t_{2}, fire(t_{1}, m)) = (m − Pre t_{1} − Pre t_{2}) +

Post t_{1} + Post t_{2}. Let us consider P, t. The functor fire t yielding a function is defined by:

(Def. 9) domfire t = ℕ^{P} and for every marking m of P holds (fire t)(m) =

fire(t, m). Next we state two propositions:

(34) rngfire t ⊆ ℕ^{P} .

(35) fire(t_{2}, fire(t_{1}, m)) = (fire t_{2} · fire t_{1})(m).

Let us consider P. A non empty set is called a Petri net over P if: (Def. 10) For every set x such that x ∈ it holds x is a transition of P.

In the sequel N denotes a Petri net over P. Let us consider P, N. We see that the element of N is a transition of P. In the sequel e, e_{1}, e_{2} denote elements of N.

4. Firing Sequences of Transitions

Let us consider P, N. A firing-sequence of N is an element of N^{∗}. In the sequel C, C_{1}, C_{2} are firing-sequences of N. Let us consider P, N, C. The functor fire C yielding a function is defined

by the condition (Def. 11). (Def. 11) There exists a function yielding finite sequence F such that fire C =

compose_{ℕ^{P}} F and lenF = lenC and for every natural number i such that i ∈ domC holds F (i) = fire(C_{i} qua element of N).

The following propositions are true:

(36) fire(ε_{N}) = id_{ℕ^{P}}.

(37) fire⟨e⟩ = fire e.

(38) fire e · id_{ℕ^{P}} = fire e.

(39) fire⟨e_{1}, e_{2}⟩ = fire e_{2} · fire e_{1}.

(40) domfire C = ℕ^{P} and rngfire C ⊆ ℕ^{P} .

(41) fire(C_{1} ^{⌢} C_{2}) = fire C_{2} · fire C_{1}.

(42) fire(C ^{⌢} ⟨e⟩) = fire e · fire C.

Let us consider P, N, C, m. The functor fire(C, m) yielding a marking of P is defined as follows:

(Def. 12) fire(C, m) = (fire C)(m).

5. Sequential Composition

Let us consider P, N. A process in N is a subset of N^{∗}.

In the sequel R, R_{1}, R_{2}, R_{3}, P_{1}, P_{2} denote processes in N.

One can verify that every function which is finite sequence-like is also finite subsequence-like.

Let us consider P, N, R_{1}, R_{2}. The functor R_{1} beforeR_{2} yields a process in N and is defined by:

(Def. 13) R_{1} beforeR_{2} = {C_{1} ^{⌢} C_{2} : C_{1} ∈ R_{1} ∧ C_{2} ∈ R_{2}}.

Let us consider P, N and let R_{1}, R_{2} be non empty processes in N. One can verify that R_{1} beforeR_{2} is non empty.

One can prove the following propositions:

(43) (R_{1} ∪ R_{2}) beforeR = (R_{1} beforeR) ∪ (R_{2} beforeR).

(44) R before(R_{1} ∪ R_{2}) = (R beforeR_{1}) ∪ (R beforeR_{2}).

(45) {C_{1}} before{C_{2}} = {C_{1} ^{⌢} C_{2}}.

(46) {C_{1}, C_{2}} before{C} = {C_{1} ^{⌢} C, C_{2} ^{⌢} C}.

(47) {C} before{C_{1}, C_{2}} = {C ^{⌢} C_{1}, C ^{⌢} C_{2}}.

6. Concurrent Composition

Let us consider P, N, R_{1}, R_{2}. The functor R_{1} concurR_{2} yielding a process in N is defined as follows:

(Def. 14) R_{1} concurR_{2} = {C : ⋁_{q_{1},q_{2} : finite subsequence} (C = q_{1} ∪ q_{2} ∧ q_{1} misses

q_{2} ∧ Seqq_{1} ∈ R_{1} ∧ Seqq_{2} ∈ R_{2})}. Let us observe that the functor R_{1} concurR_{2} is commutative.

Next we state four propositions:

(48) (R_{1} ∪ R_{2}) concurR = (R_{1} concurR) ∪ (R_{2} concurR).

(49) {⟨e_{1}⟩} concur{⟨e_{2}⟩} = {⟨e_{1}, e_{2}⟩, ⟨e_{2}, e_{1}⟩}.

(50) {⟨e_{1}⟩, ⟨e_{2}⟩} concur{⟨e⟩} = {⟨e_{1}, e⟩, ⟨e_{2}, e⟩, ⟨e, e_{1}⟩, ⟨e, e_{2}⟩}.

(51) (R_{1} beforeR_{2}) beforeR_{3} = R_{1} before(R_{2} beforeR_{3}).

Let p be a finite subsequence and let i be a natural number. The functor Shift^{i} p yielding a finite subsequence is defined as follows:

(Def. 15) domShift^{i} p = {i + k; k ranges over natural numbers: k ∈ domp} and for

every natural number j such that j ∈ domp holds (Shift^{i} p)(i + j) = p(j). In the sequel q, q_{1}, q_{2} denote finite subsequences. One can prove the following propositions:

(52) Shift^{0} q = q.

(53) Shift^{i+j} q = Shift^{i} Shift^{j} q.

(54) For every finite sequence p such that p ≠ ∅ holds domShift^{i} p = {j_{1} : i + 1 j_{1} ∧ j_{1} i + lenp}.

(55) For every finite subsequence q holds q = ∅ iff Shift^{i} q = ∅.

(56) Let q be a finite subsequence. Then there exists a finite subsequence s_{1} such that doms_{1} = domq and rngs_{1} = domShift^{i} q and for every k such that k ∈ domq holds s_{1}(k) = i + k and s_{1} is one-to-one.

(57) For every finite subsequence q holds \overline{\overline{q}} = \overline{\overline{Shift^{i} q}}.

(58) For every finite sequence p holds domp = domSeqShift^{i} p.

(59) For every finite sequence p such that k ∈ domp holds (SgmdomShift^{i} p)(k) = i + k.

(60) For every finite sequence p such that k ∈ domp holds (SeqShift^{i} p)(k) = p(k).

(61) For every finite sequence p holds SeqShift^{i} p = p.

In the sequel p_{1}, p_{2} are finite sequences.

One can prove the following propositions:

(62) dom(p_{1} ∪ Shift^{len p_{1}} p_{2}) = Seg(lenp_{1} + lenp_{2}).

(63) For every finite sequence p_{1} and for every finite subsequence p_{2} such that lenp_{1} i holds domp_{1} misses domShift^{i} p_{2}.

(64) For all finite sequences p_{1}, p_{2} holds p_{1} ^{⌢} p_{2} = p_{1} ∪ Shift^{len p_{1}} p_{2}.

(65) For every finite sequence p_{1} and for every finite subsequence p_{2} such that i lenp_{1} holds p_{1} misses Shift^{i} p_{2}.

(66) (R_{1} concurR_{2}) concurR_{3} = R_{1} concur(R_{2} concurR_{3}).

(67) R_{1} beforeR_{2} ⊆ R_{1} concurR_{2}.

(68) If R_{1} ⊆ P_{1} and R_{2} ⊆ P_{2}, then R_{1} beforeR_{2} ⊆ P_{1} beforeP_{2}.

(69) If R_{1} ⊆ P_{1} and R_{2} ⊆ P_{2}, then R_{1} concurR_{2} ⊆ P_{1} concurP_{2}.

(70) For all finite subsequences p, q such that q ⊆ p holds Shift^{i} q ⊆ Shift^{i} p.

(71) For all finite sequences p_{1}, p_{2} holds Shift^{len p_{1}} p_{2} ⊆ p_{1} ^{⌢} p_{2}.

(72) If domq_{1} misses domq_{2}, then domShift^{i} q_{1} misses domShift^{i} q_{2}.

(73) For all finite subsequences q, q_{1}, q_{2} such that q = q_{1} ∪ q_{2} and q_{1} misses q_{2} holds Shift^{i} q_{1} ∪ Shift^{i} q_{2} = Shift^{i} q.

(74) For every finite subsequence q holds domSeqq = domSeqShift^{i} q.

(75) For every finite subsequence q such that k ∈ domSeqq there exists j such that j = (Sgmdomq)(k) and (SgmdomShift^{i} q)(k) = i + j.

(76) For every finite subsequence q such that k ∈ domSeqq holds (SeqShift^{i} q)(k) = (Seqq)(k).

(77) For every finite subsequence q holds Seqq = SeqShift^{i} q.

(78) For every finite subsequence q such that domq ⊆ Segk holds domShift^{i} q ⊆ Seg(i + k).

(79) Let p be a finite sequence and q_{1}, q_{2} be finite subsequences. If q_{1} ⊆ p, then there exists a finite subsequence s_{1} such that s_{1} = q_{1} ∪ Shift^{len p} q_{2}.

(80) Let p_{1}, p_{2} be finite sequences and q_{1}, q_{2} be finite subsequences. Suppose q_{1} ⊆ p_{1} and q_{2} ⊆ p_{2}. Then there exists a finite subsequence s_{1} such that s_{1} = q_{1} ∪ Shift^{len p_{1}} q_{2} and domSeqs_{1} = Seg(lenSeqq_{1} + lenSeqq_{2}).

(81) Let p_{1}, p_{2} be finite sequences and q_{1}, q_{2} be finite subsequences. Suppose q_{1} ⊆ p_{1} and q_{2} ⊆ p_{2}. Then there exists a finite subsequence s_{1} such that s_{1} = q_{1} ∪ Shift^{len p_{1}} q_{2} and domSeqs_{1} = Seg(lenSeqq_{1} + lenSeqq_{2}) and Seqs_{1} = Seqq_{1} ∪ Shift^{len Seq q_{1}} Seqq_{2}.

(82) Let p_{1}, p_{2} be finite sequences and q_{1}, q_{2} be finite subsequences. Suppose q_{1} ⊆ p_{1} and q_{2} ⊆ p_{2}. Then there exists a finite subsequence s_{1} such that s_{1} = q_{1} ∪ Shift^{len p_{1}} q_{2} and (Seqq_{1})^{⌢} (Seqq_{2}) = Seqs_{1}.

(83) (R_{1} concurR_{2}) before(P_{1} concurP_{2}) ⊆ (R_{1} beforeP_{1}) concur(R_{2} beforeP_{2}). Let us consider P, N and let R_{1}, R_{2} be non empty processes in N. Note

that R_{1} concurR_{2} is non empty.

7. Neutral Process

Let us consider P and let N be a Petri net over P. The neutral process in N yields a non empty process in N and is defined as follows:

(Def. 16) The neutral process in N = {ε_{N}}.

Let us consider P, let N be a Petri net over P, and let t be an element of N. The elementary process with t yielding a non empty process in N is defined as follows:

(Def. 17) The elementary process with t = {⟨t⟩}.

One can prove the following propositions:

(84) (The neutral process in N) beforeR = R.

132 grzegorz bancerek et al.

(85) R beforethe neutral process in N = R.

(86) (The neutral process in N) concurR = R.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.

[5] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[8] Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar - part 1. Formalized Mathematics, 2(5):683–687, 1991.

[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[11] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

[12] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[13] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[14] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[15] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[16] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[18] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Contents Formaliz. Math. 11 (1)

Subspaces and Cosets of Subspace of Real Unitary Space

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Operations on Subspaces in Real Unitary Space

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Linear Combinations in Real Unitary Space

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Dimension of Real Unitary Space

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Trigonometric Functions on Complex Space

By Takashi Mitsuishi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Topology of Real Unitary Space

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Armstrong’s Axioms

By William W. Armstrong et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Convex Sets and Convex Combinations

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Quotient Vector Spaces and Functionals

By Jarosław Kotowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bilinear Functionals in Vector Spaces

By Jarosław Kotowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Hermitan Functionals. Canonical Construction of Scalar Product

in Quotient Vector Space

By Jarosław Kotowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Continued on inside back cover

The Class of Series-Parallel Graphs. Part I

By Krzysztof Retel  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Term Orders

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Polynomial Reduction

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Processes in Petri Nets

By Grzegorz Bancerek et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125